Biased Coin Toss Probability Calculator

Alice and Bob want to choose between the opera and the movies by tossing a fair coin. Write a program that simulates coin tossing. @9: The weaker team has the same probability as the stronger team of winning the coin toss, assuming a fair (p=0. Difference is in how often I change coin bias (random 0<= p_Heads <=1):. So these two facts here, multiplication rule and the total probability theorem, are basic tools that one uses to break down probability calculations into a simpler parts. Let us learn more about coin toss probability formula. Example: Telephone sampling is common in marketing surveys. f = number of times head appears. Consider 10 independent tosses of a biased coin with the probability of Heads at each toss equal to p, where 0. Do not enter anything in the column for odds. An unfair coin with P(H)=0. This binomial distribution calculator is here to help you with probability problems in the following form: what is the probability of a certain number of successes in a sequence of events? Read on to learn what exactly is the binomial probability distribution, when and how to apply it, as well as to learn the binomial probability formula. A fair coin is tossed 5 times. Suppose a coin is tossed 170 times and you observed that 50 heads and 120 tails. A coin is sampled at random from the box (this means that each of the three has the same probability of being drawn, and that the draw of the coin is independent of its subsequent flip), and then it is flipped. privately and independently conducts one toss of a biased coin, with unknown probability wH of heads. Spotting the difference between real and imaginary coin toss results is incredibly easy. Consider a sequence of independent tosses of a biased coin at times k=0,1,2,…,n. Suppose I have an unfair coin, and the probability of flip a head (H) is p, probability of flip a tail (T) is (1-p). A probability of one means that the event is certain. The following example shows how a sample can be biased, even though there is some randomness in the selection of the sample. Toss results can be viewed as a list of individual outcomes, ratios, or table. This example is not only for relative frequency, but it also clears that during random experiment we mostly took the probability of head ½. Toss a fair coin twice. When calculated, the probability of this happening is 1/1024 which is about 0. Your function should use only foo(), no other library method. Spotting the difference between real and imaginary coin toss results is incredibly easy. Since the random function generates uniform distribution, I feels that the above code is good enough for simulating a probability event. " In contrast to the more familiar "probability" that represents the uncertainty that a single outcome will occur, "entropy" quantifies the aggregate uncertainty of all possible outcomes. 5 for coin A and q=0. Consider, you toss a coin once, the chance of occurring a head is 1 and chance of occurring a tail is 1. So we find probabilities of. Tossing a Biased Coin Michael Mitzenmacher When we talk about a coin toss, we think of it as unbiased: with probability one-halfit comes up heads, and with probability one-halfit comes up tails. 15 Binomial Theorem 3 Suppose a shipment has 5 good items and 2. and the probability of switching between them: a coin-toss. Note that it does NOT say that the coin is bias. I would like to estimate the probability of getting at least 1 “tail” when I toss a fair coin 3 times. For example, for the correct coin toss analogy you need to have two coins, one of which is a regular coin with a heads and a tails side. The probability for an unbiased coin (defined for this purpose as one whose probability of coming down heads is somewhere between 45% and 55%) is small when compared with the alternative hypothesis (a biased coin). Dice Roll Probability for 6 Sided Dice: Sample Spaces. Predicting a coin toss. When the coin lands, that party is winner whose chosen side. That was flip number Flip again? Color The Coin! Share The Coin! Facebook Twitter WhatsApp. Essentially, as $\alpha$ becomes larger the bulk of the probability distribution moves towards the right (a coin biased to come up heads more often), whereas an increase in $\beta$ moves the distribution towards the left (a coin biased to come up tails more often). The variance of the binomial distribution is: σ 2 = Nπ(1-π) where σ 2 is the variance of the binomial distribution. I have tried to address this on a similar question somewhere. In simple situations, intuition is a reasonable guide. Theoretical Probability: probability based on reasoning written as a ratio of the number of favorable outcomes to the number of possible outcomes. What is the conditional probability that. If we have a biased coin, i. When a coin is tossed, there lie two possible outcomes i. the probability of a head is greater than 0. , Probability of getting a trial is the same each time we toss the coin 17 Binomial Distribution: Example 1 Let’s say that we toss a coin n (=100) times It is a biased coin; the chance of Head is 0. If this is a formula for the expected number of lead changes with a fair coin (50% heads) as discussed in this thread, it is clearly wrong. What is the probability of getting (i) three heads, (ii) two heads, (iii) one head, (iv) 0 head. A probability of one means that the event is certain. The Binomial Distribution. We can specify each possible outcome in advance - heads or tails. Because the coin toss is the simplest random event you can imagine, many questions about coin tossing can be asked and answered in great depth. [5 points] Suppose we have a biased coin that comes up heads with probability pfor some 0. In the case of an unbiased coin, the probability that the toss will result in a heads is the same probability that it will be a tails, 0. So, we’ll use one-tailed probability. they adjust the weight of the coins in such a way that the one side of the coin is more likely than the other while tossing. Suppose that one of these three coins is selected at random and °ipped. In cricket, the coin toss is used to decide which team will go for batting. A Bayesian sees one particular experiment and uses this to test some hypothesis. Statistical distance between uniform and biased coin independent and each bit is $1$ with probability $1/2-\epsilon$. How to use probability in a sentence. Intuitively, it seems that introducing a small bias (p = 0. The simplicity of the coin toss also opens the road to more advanced probability theories dealing with events with an infinite number of possible outcomes. And they probably also know that coins are not perfectly balanced. 75; Same as before, in the first case, there are 50–50 chances of heads or tails. The notation for conditional probability is P(B|A) [pronounced as The probability of event B given A]. 5 or 50% chance of the outcome occurring. 42 There are three coins in a box. I think this is a refreshing approach to an age-old problem- the coin toss. Game Theory (Part 11) John Baez. We toss two fair coins simultaneously and independently. Bayesian Inference. Find the probability of obtaining a) two tails on first two tosses b) a head, a tail and a head( in that order) c)two heads and one tail (in any prder) ???. The point estimate refers to the probability of getting one of the results. Now this seems like it's impossible. Problem 4 [30 pts, (3,12,3,12. Using a prior. Alice and Bob want to choose between the opera and the movies by tossing a fair coin. Then this paper will analyse the coin flip probabilistically, also considering the factor of human bias. The QuantWolf Coin Toss Runs Calculator will calculate the mean given probability of heads and run length. To calculate the probability of an event A when all outcomes in the sample space are equally likely, count the number of outcomes for event A and divide by the total number of outcomes in the sample space. A common topic in introductory probability is solving problems involving coin flips. To send the entire sequence will require one million bits. So for four tosses. Apart from bias, there’s a second component to the generalization error, consisting of the variance of a model fitting procedure. Consider 10 independent tosses of a biased coin with the probability of Heads at each toss equal to p, where 0 < p < 1. It's a coin that results in heads with probability p. For example, for the correct coin toss analogy you need to have two coins, one of which is a regular coin with a heads and a tails side. For examples: If you toss a fair dime and a fair nickel, you will see four possible outcomes. A common topic in introductory probability is solving problems involving coin flips. A fair coin is tossed two times. In the “die-toss” example, the probability of event A, three dots showing, is P(A) = 1 6 on a single toss. This is very similar to Q1, the only difference is that in this case the coin is biased. How to create an unfair coin and prove it with math The obvious way to calculate this probability is simply to divide the number of heads by the total number of. If the coin is tossed 4 times, what is the probability of getting a. you'll lose 50 cents if the other coin shows tail. 1 shows the results of tossing a coin 5000 times twice. to be introduced in the next section, we shall be able to prove the Law of Large Numbers. After all, real life is rarely fair. probability of B is the probability of this, which is A intersection B, plus the probability of that, which is A complement intersection B. This theorem will justify mathematically both our frequency concept of probability and the interpretation of expected value as the average value to be expected in a large number of experiments. Example: You asked your 3 friends Shakshi, Shreya and Ravi to toss a fair coin 15 times each in a row and the outcome of this experiment is given as below:. After you have tossed your biased coin for a certain number of times and you've collected enough data pertaining to the "behavior" of the coin, you can use that data when using the point estimate calculator. In fact, the probability for most other values virtually disappeared — including the probability of the coin being fair (Bias = 0. Imagine flipping a coin 1000 times, and counting the number of heads. A biased coin (with probability of obtaining a Head equal to p > 0) is tossed repeatedly and independently until the first head is observed. Tossing Coins Experiment and set the conditions to 2 coins and 40 throws. The probability of that occurring is the same as the probability of getting a heads on every toss because the probability of getting a heads or tails on any one toss is 50 percent. On the other hand, if we had a perfect coin with half-half chances of coming up heads or tails upon a coin toss, then we can guess the outcome of a toss with only 50% accuracy (probability 0. Coin Toss: Simulation of a coin toss allowing the user to input the number of flips. Let us take the experiment of tossing two coins simultaneously:. b) What do you think E[X] should be. If it's a fair coin, the two possible outcomes, heads and tails, occur with equal probability. 15 Binomial Theorem 3 Suppose a shipment has 5 good items and 2. There is a 50% chance of showing heads and a 50% chance of showing tails. Since tossing a coin is random, the coin should not alternate between heads and tails. To finish the example, you would divide five by 36 to find the probability to be 0. Problem: A coin is biased so that it has 60% chance of landing on heads. For each toss of the coin the program should print Heads or Tails. The sample space. So the standard. Flip a coin a bunch of times and observe the results, e. The two events are (1) first toss is a head and (2) second toss is a head. This probability does not describe the short-term results of an experiment. What is the probability of getting (i) all heads, (ii) two heads, (iii) at least one head, (iv) at least two heads?. This could depend on many things that aren't stated in the question. "The coin tosses are independent events; the coin doesn't have a memory. That seems highly improbable. To start to work out the solution to the problem, I will set k to a value of three - in other words, we will be trying to see what is the probability of seeing three consecutive heads is at toss i, given that there have not been three heads at an earlier toss. guess the bias of the coin (heads-biased or tails-biased) according to which of these two hypotheses has the larger nal subjective probability. Indeed, even if we have strong empirical evidence that the coin is biased towards heads with probability, say, 0. The event h,t or t,h are equi-likely, without any bias we can call that if Event h,t occurs it means head, t,h means tails but if h,h or t,t occurs we repeat the experiment. $ Ac is none of coin tosses comes up heads, and P(A) = 1 − P(Ac). Using a prior. 5 (50%) for either of the two possible results. Predicting a coin toss. Use “first step analysis” to write three equations in three unknowns (with two additional boundary conditions) that give the expected duration of the game that the gambler plays. When considering only a finite number of throws, the probability of making money can be calculated exactly. What is the probability of getting (i) three heads, (ii) two heads, (iii) one head, (iv) 0 head. gl/2z3jX6 In this video you will learn how to find Probability given that Coin Toss may be Unfair. This is a very simple model, yet surprisingly powerful. This is fun for conspiracy theorists, but is of course nonsense - hence why the Super Bowl coin toss odds are always the same and always equal. define a function that creates a biased coin (which is a function), given the original coin, the bias. P(16 heads) = 0. If this probability is very low, we might make the inference that the coin is biased. In fact, the probability for most other values virtually disappeared — including the probability of the coin being fair (Bias = 0. On the other hand, the situation might change if one learns that the coin has been altered so heads are more likely. The more times one flips a coin, the more closely one approaches the theoretical 50 percent. To learn more about the binomial distribution, go to Stat Trek's tutorial on the binomial distribution. The gambler's fallacy can be illustrated by considering the repeated toss of a fair coin. Let B be the event that the 9th toss results in. 75; Same as before, in the first case, there are 50–50 chances of heads or tails. Because the coin toss is the simplest random event you can imagine, many questions about coin tossing can be asked and answered in great depth. Guessing the probability of heads tossing two biased coins (self. Possible Outcomes Calculator. Interview question for Strategist in New York, NY. If we toss a coin, assuming that the coin is fair, then heads and tails are equally likely to appear. To illustrate this idea, we will use the Binomial distribution, B(x; p), where p is the probability of an event (e. Here's a game. Pae and Loui [5] further proved that the mapping function used by Elias is optimal among all -randomizing functions and is computable in polynomial time. Although I have to say that almost every person I have ever known to do a coin-toss uses the palm invert trick to add that element of "manual randomness". Tossing Coins Experiment and set the conditions to 2 coins and 40 throws. The expected value is found by multiplying each outcome by its probability and summing. We need the height of the cone what we do is we know that when height is 10, V=something, so solve for the diameter is at that point and compare it to 24, the full height sorry if confusing V=(1/3)hpir^2 when h=10 160pi/3=1/3(10)pir^2 ties 3 both sides 160pi=10pir^2 divide by 10pi 16=r^2 sqrt both sides r=4 at that point, the radius is 4 the real radius of the ful is 24/2=12 4/12=1/3 it is 1/3. altering a coin can be done by wrapping its function into another function that alters its result by making use of some random to override the original output with a new given probability. Probability is the measurement of chances – likelihood that an event will occur. You should also not enter anything for the answer, P(H|D). Think of a trial of any experiment, for example, tossing a coin. Sample Spaces and Random Variables: examples. Online virtual coin toss simulation app. and allow us to prescribe criteria for designing coins with a prescribed probability distribution of landing on heads, tails, or sides. ] (b) Now consider two coins, of which one is normal and the other is biased so that the probability of obtaining a Head is p>1/2. Suppose that a coin has the characteristics that when flipped, the probability of getting a head is. Conceptually in a unbiased or fair coin both the sides have the same probability of showing up i. Probability is the measurement of chances – likelihood that an event will occur. [You will need to use the Binomial distribution from M1S. 50 or 50 % probability exactly when experimented with both sides alternately. After flipping heads, say, five consecutive times, our inclination is to predict an increase in likelihood that the next coin toss will be tails — that the. Find the probability of obtaining a) two tails on first two tosses b) a head, a tail and a head( in that order) c)two heads and one tail (in any prder) ???. As you learned in Chapter 3, if you toss a fair coin, the probability that the result is heads is 0. For anyone taking first steps in data science, Probability is a must know concept. So for four tosses. If the coin flips are independent and if each toss results in heads with probability [math]p\in(0,1)[/math] then the probability of "matching" a pair of flips is given by [math]p^2+(1-p)^2[/math]. 4 it is argued you can’t easily produce a coin that is biased when flipped (and caught). These 4 outcomes will form a sample space. 5 for each side. For each toss of the coin the program should print Heads or Tails. 5), and the alternative hypothesis is Ha: the coin is biased in favor of a head (i. Let us understand the concept of experimental probability through examples. of choosing two coins reduces to the four outcomes of choosing either the fair coin (1), or a biased coin (2 or 3) for each flip. Alice spins a coin on a table and waits for it to land on one side. Conditional probability: 2009-06-01: From Tanja: A biased coin where P(heads) =3/5 is flipped 4 times. Probabilities are usually given as percentages. 1 Given a. As you learned in Chapter 3, if you toss a fair coin, the probability that the result is heads is 0. IOn order to ascertain which type of coin we have, we shall perform the following statistical test: We shall toss the coin 1000 times. Imagine you toss a fair coin 20 times. Let us understand the concept of experimental probability through examples. A biased coin is tossed 6 times. coin toss probability calculator,monte carlo coin toss trials. Individual coin tosses are described by the Bernoulli distribution with parameter x, the latent variable or bias of the coin. describing the probability of observing the data, and (iii) a criterion that allows us to move from the data and model to an estimate of the parameters of the model. coin toss probability calculator,monte carlo coin toss trials. Probability that we get outcome in 1st event is 2x(1-x). Here are some examples. Small probabilities add up when chances are taken repeatedly. This is fun for conspiracy theorists, but is of course nonsense - hence why the Super Bowl coin toss odds are always the same and always equal. Similarly, if a player is at one consecutive head so far on any toss, the probability that they will be at two consecutive heads after the next toss is 50% and the probability of dropping back to. The outcomes in different tosses are statistically independent and the probability of getting heads on a single toss is 1 / 2 (one in two). A Bayesian sees one particular experiment and uses this to test some hypothesis. On a biased coin the probability of it showing heads for a given coin toss is 0. Every flip of the coin has an “independent probability“, meaning that the probability that the coin will come up heads or tails is only affected by the toss of the coin itself. With R we can play games of chance - say, rolling a die or flipping a coin. If you want to guess how many times you will get heads if you flip a fair coin 10 times you might assign the value. For K-12 kids, teachers and parents. The number cube and the coin are fair. The pyramid is then reflected over the xz-plane. If we select a fair coin the probability of heads is 1/2. 7% includes the probability of throwing 527, 528, 529. probability of any continuous interval is given by p(a ≤ X ≤ b) = ∫f(x) dx =Area under f(X) from a to b b a That is, the probability of an interval is the same as the area cut off by that interval under the curve for the probability densities, when the random variable is continuous and the total area is equal to 1. A fair coin is tossed two times. When the coin lands, that party is winner whose chosen side. In this lab, we are going to look at basic probability and how to conduct basic simulations using R. Let A be the event that there are 6 Heads in the first 8 tosses. (Image: carnival image by Janet Wall from Fotolia. It is one trial of a binomial distribution. PART A: Find the probability of X. $\endgroup$ - TinaW Nov 5 '16 at 18:59. The variance of the number of heads is 1000*(1/2)*(1/2)=250. For a coin toss: E(Heads)= 0*(0. The obverse (principal side) of a coin typically features a symbol intended to be evocative of stately power, such as the head of a monarch or well-known state representative. For example, a coin toss, the role of a die, and the dealing of playing cards. If so, we shall call the outcome heads; if not we call. Last time we talked about independence of a pair of outcomes, but we can easily go on and talk about independence of a longer sequence of outcomes. "On average", we would expect to get 500 heads. Let the program toss the coin 100 times, and count the number of times each side of the coin appears. Now flip them both at the same time. We can specify each possible outcome in advance - heads or tails. There are a large number of probability distributions available, but we only look at a few. Carnival game operators use probability formulas to make winning difficult. Recall that 210 = 1024. Suppose I have an unfair coin, and the probability of flip a head (H) is p, probability of flip a tail (T) is (1-p). So for our biased coin toss the expected value is P(0) * 0 + P(1) * 1 = (1 - x) * 0 + x * 1 =. Information Theory and Statistics. This book is very mathematical. Solution:. Write a R code to calculate the probability. That seems highly improbable. We cannot examine the possibility that the coin was unfair because the probability of getting heads or tails is unknown for a biased coin. The conditional probability of an event B in relationship to an event A is the probability that event B occurs given that event A has already occurred. Example 2: Coin-A is tossed 200 times, and the relative occurrence of Tails is 0. And that in no way affects the probability of getting a tails on the third flip. If the gambler has $1 he plays with a coin that gives probability p = 3 ∕ 4 of winning a dollar and probability q = 1 ∕ 4 of losing a dollar. Rakhshan and H. The Null Hypothesis is the theory we can test directly. On Information and Sufficiency. of choosing two coins reduces to the four outcomes of choosing either the fair coin (1), or a biased coin (2 or 3) for each flip. Try to find the expected number of coin toss that would be required to call heads or tails? Let expected coin toss be e. Let B be the event that the 9th toss results in. Word problems on coin toss probability: 1. Then this paper will analyse the coin flip probabilistically, also considering the factor of human bias. This book is very mathematical. Divide the number of ways to achieve the desired outcome by the number of total possible outcomes to calculate the weighted probability. The program should call a separate function flip()that takes no arguments and returns 0 for tails and 1 for heads. We look at some of the basic operations associated with probability distributions. A biased coin (with probability of obtaining a Head equal to p > 0) is tossed repeatedly and independently until the first head is observed. Let’s consider a ‘success’ to be when heads appears in the coin toss. Let p 0 denote the prior probability of the coin being biased to heads. However, it is not small enough to cause us to believe that the coin has a significant bias. Assuming equal probabilities for girl and boy births, you could simulate the births in three-child families by tossing three fair coins and observing the outcomes—tails for boys and heads for girls. Let's write a function that takes in two arguments: 1. When many of us think of probabilities, the first thing that comes to mind is a coin toss—having a 50% chance at being right on a given toss. Pr[H]= p 1, Pr[T]= p 2. Coin Toss Probability Calculator Coin toss also known as coin flipping probability is used by people around the world to judge whether its going to be head or tail after flipping the coin. This is what i have so farI need to add a function named coin to simulate a coin toss where heads is represented by a 1 and tails a 2. We will always see Head, so probability of getting Head with a biased coin=1. Consider a sequence of independent tosses of a biased coin at times k=0,1,2,…,n. A Bayesian sees one particular experiment and uses this to test some hypothesis. Byju's Coin Toss Probability Calculator is a tool which makes calculations very simple and interesting. The game, called Penney Ante, involves flipping a coin, which you assume has equal probability of coming up heads or tails. The expected value is found by multiplying each outcome by its probability and summing. On each toss, the probability of Heads is p, and the probability of Tails is 1−p. The probability of tossing 14 or more heads out of 20 is: See Binomial Distribution. Let’s assume that there is a company that makes biased coins i. 7870 and the probability of getting three or more heads in a row or three or more tails in a row is 0. Bayesian Inference. one another, we can multiply these probabilities: the probability of all n balls not going into the. The outcomes in different tosses are statistically independent and the probability of getting heads on a single toss is 1 / 2 (one in two). it is empty, is 1 − 1 n: m. Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Coin Tossing Project I. 1 Analysis versus Computer Simulation A computer simulation is a computer program which attempts to represent the real world based on a model. A Drug Influence Evaluation (DIE) is a formal assessment of an impaired driving suspect, performed by a trained law enforcement officer who uses circumstantial facts, questioning, searching, and a physical exam to form an unstandardized opinion as to whether a suspect’s driving was impaired by drugs. Probability-and-statistics-> SOLUTION: a biased coin has 1 in 10 chance of landing heads. Tack-tossing is a super-easy experiment. If the probability of an event is high, it is more likely that the event will happen. 2The Axioms of Probability37Now suppose that we toss a coin three times but we count the number of heads in threetosses instead of. In the case of an unbiased coin, the probability that the toss will result in a heads is the same probability that it will be a tails, 0. For anyone taking first steps in data science, Probability is a must know concept. Imagine doing this experiment 9000(f+ b) times. A calculator is provided here to show the probability of avoiding a danger given the probability and the number of repetitions of the risk. What is the probability of getting (i) three heads, (ii) two heads, (iii) one head, (iv) 0 head. Imagine flipping a coin 1000 times, and counting the number of heads. 50, we can take the binomial probabilities from Table B in Appendix D: P(14 heads) = 0. gl/2z3jX6 In this video you will learn how to find Probability given that Coin Toss may be Unfair. As you learned in Chapter 3, if you toss a fair coin, the probability that the result is heads is 0. If one additional student earns a 100% in the class, what is the new class average. The notation for conditional probability is P(B|A) [pronounced as The probability of event B given A]. altering a coin can be done by wrapping its function into another function that alters its result by making use of some random to override the original output with a new given probability. a fair coin outputs 0 with a probability of 0. Assuming equal probabilities for girl and boy births, you could simulate the births in three-child families by tossing three fair coins and observing the outcomes—tails for boys and heads for girls. We’ll imagine that the universe is entirely causal (every event has a cause), and talk about the two different ways (epistemiv vs. In the case of the coin toss, the Null Hypothesis would be that the coin is fair, and has a 50% chance of landing as heads or tails for each toss of. # 9: Rejection Sampling and Monte-Carlo Method-1. A number cube is rolled and a coin is tossed. After all, real life is rarely fair. SOLUTION: Define:. Setting the response distribution to 50% is the most conservative assumption. If the outcomes of the two coin tosses are the same, we win; otherwise, we lose. To determine if a coin flip is truly random, this paper will first analyse the interactions of a coin flip through classical mechanics, and then apply the ideas of quantum mechanics, chaos theory, and implicit human bias. If the result is heads, Alice wins $1 from Bob; if tails, Alice pays $1 to Bob. is small when compared with the alternative hypothesis (a biased coin). We can explore this problem with a simple function in python. 6 and that. b) lands heads at least three times c) lands tails at most one time. In cricket, the coin toss is used to decide which team will go for batting. Statistical distance between uniform and biased coin independent and each bit is $1$ with probability $1/2-\epsilon$. Simulating a coin toss in excel I guess when you start to look at gambling theories or probabilities the natural place to start is the coin toss. [5 points] Suppose we have a biased coin that comes up heads with probability pfor some 0 > > import LongestHeadRun as lhr # Find number of sequences of length 8 where the longest run of heads is 3 for # a fair coin >> > lhr. However, it is not small enough to cause us to believe that the coin has a significant bias. The original question was: Recently I've come across a task to calculate the probability that a run of at least K successes occurs in a series of N (K≤N) Bernoulli trials (weighted coin flips), i. Probability, physics, and the coin toss L. For additional details, including an interactive probability calculator, please visit the z Score Probability Calculator. The other has heads on both sides. A number cube is rolled and a coin is tossed.