Swann Dive, a systems programmer at Poly Unsaturated,...

Swann Dive, a systems programmer at Poly Unsaturated, offered his friend Charlie Tuna, an application programmer, the following proposition. On each roll of three dice Swann would pay Charlie one dollar for each ace that showed; if no aces were turned Charlie would pay Swann one dollar. Charlie reasons that the probability of rolling an ace on the first die is $\frac{1}{6}$; similarly for the second and third die. Hence the probability $3 \times \frac{1}{6}=\frac{1}{2}$ of getting at least one ace and he might get two or even three of them. (a) Is Charlie right-that is, is it a good proposition for him? (b) What is the probability that Charlie will roll one, two, or three aces, respectively? (c) What is the average amount of money Charlie can expect to win each time the dice are rolled? (Swann didn't tell him, but this game is known as chuck-a-luck at carnivals.)

Swann Dive, a systems programmer at Poly Unsaturated, offered his friend Charlie Tuna, an application programmer, the following proposition. On each roll of three dice Swann would pay Charlie one dollar for each ace that showed; if no aces were turned Charlie would pay Swann one dollar. Charlie reasons that the probability of rolling an ace on the first die is $\frac{1}{6}$; similarly for the second and third die. Hence the probability $3 \times \frac{1}{6}=\frac{1}{2}$ of getting at least one ace and he might get two or even three of them.
(a) Is Charlie right-that is, is it a good proposition for him?
(b) What is the probability that Charlie will roll one, two, or three aces, respectively?
(c) What is the average amount of money Charlie can expect to win each time the dice are rolled? (Swann didn't tell him, but this game is known as chuck-a-luck at carnivals.)

Key Concepts

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, with each trial having the same probability of success. When rolling dice, each die can be seen as a trial with a probability of landing on a designated face (such as an ace), and the total number of successes (aces rolled) follows a binomial pattern.

Expected Value

The expected value is a statistical measure that represents the average outcome of a random process over a large number of trials. In the context of games or bets, it is calculated by multiplying each potential outcome by its probability and summing these products, which indicates whether a game is favorable or unfavorable to the players.

Independence of Events

Independence means that the outcome of one trial does not affect the outcomes of others. For dice rolls, this principle ensures that the result of one die does not influence the result of another, which is crucial for correctly applying probability models like the binomial distribution.

Combinatorial Analysis

Combinatorial analysis involves counting the number of ways certain outcomes can occur, such as obtaining a specific number of aces in a set of dice rolls. This approach uses combinatorial techniques, including combinations, to determine the probabilities of various outcomes by comparing the number of favorable cases to the total number of cases.

Transcript

00:01 The game described in this exercise is basically one where we have a random number of chips which is defined by this probability distribution.

00:09 And each of the chips has a random number amount of winnings, which is defined by this probability distribution.

00:21 So the total winnings is a function of both the number of chips and the amount one by each chip.

00:27 So in this question we are asked to create a program that simulates the total winnings when this game is played.

00:34 So i have produced a program in our code.

00:46 So first i would just point out that we have two probability distributions, so there's that one, which corresponds to the number of chips, and there's this one, which is the amount one by each chip.

01:01 And then for each simulated game, we first draw a random number of chips, which is drawn from this distribution, and then we use this number to determine the sample size that we draw from our winnings.

01:19 So if we happen to end up with 10 chips randomly, then we would draw 10 values from this probability distribution to represent the amount that is won by those 10 chips.

01:34 And so this will provide us with a vector that contains the winnings for each of the chips.

01:39 And then to find the total winnings for that game, we just sum the winnings from each chip.

01:44 And we store the total winnings for the jth simulation in a vector called winnings, which we begin the program by setting to null.

01:55 So what we end up with is a vector called winnings, that is 10 ,000 in size, and each entry corresponds to the winnings of a simulated game.

02:09 In part a, we were asked what the probability is that somebody playing the game will win more than $11 ,000.

02:19 So in r we can simply type, and this will provide us with that probability.

02:41 Running this in r, i get 0 .3031.

02:48 And for b, we are looking for the expected winnings.

02:53 So the r code for that is simply the mean of the winnings.

03:06 And in my simulation, i got $8 ,831 .131...

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