Following is a description of the game of craps. A player...

Following is a description of the game of craps. A player rolls two dice and computes the total of the spots showing. If the player's first toss is a 7 or an 11 , the player wins the game. If the first toss is a $2,3,$ or $12,$ the player loses the game. If the player rolls anything else $(4,5,6,8,9 \text { or } 10)$ on the first toss, that value becomes the player's point. If the player does not win or lose on the first toss, he tosses the dice repeatedly until he obtains either his point or a $7 .$ He wins if he tosses his point before tossing a 7 and loses if he tosses a 7 before his point. What is the probability that the player wins a game of craps? [Hint: Recall Exercise 2.119.]

Key Concepts

Conditional Probability

Conditional probability is the likelihood of an event occurring given that another event has already taken place. In probability problems where the outcome depends on a preceding event, this concept ensures the correct adjustment of probabilities based on the known conditions.

Sample Space

The sample space refers to the set of all possible outcomes of a random experiment. In the context of rolling dice, it involves enumerating every combination of the faces that can appear, and it forms the basis for calculating probabilities by assigning likelihoods to each outcome.

Probability Distribution of Dice Sums

This concept involves determining how the sums of two dice are distributed, recognizing that some sums occur more frequently than others due to the number of ways they can be obtained. Understanding this distribution is essential for calculating the probabilities of various outcomes in games involving dice.

Total Probability Theorem

The Total Probability Theorem is a rule that relates marginal probabilities to conditional probabilities. It provides a framework for evaluating the overall probability of an event by considering all possible scenarios that lead to that event, each weighted by their respective probabilities.

Markov Processes

Markov processes describe systems where the future state depends only on the current state and not on the sequence of events that preceded it. This memoryless property is useful in analyzing games or processes that involve repeating steps until a particular condition or absorbing state is reached.

Transcript

00:02 All right, we want to know what the probability is of a win in craps.

00:08 So we're trying to figure out what is the probability of a win.

00:14 Okay, so we need to figure out a couple of things, like what is the different probabilities based on these role situations that we're given? and we know that the probability of rolling a 7 is going to be equal to 6 out of 36.

00:33 Because there's six ways that you can roll a seven, right? a one and a six, one a six, a six and a one, five, and a two, two and a five, four, and a three, three, and a four, right? so there's seven, sorry, six ways to roll seven.

00:54 There's also only two ways to roll an eleven, right? so that's going to be a six and a five, right? right, or a five and a six.

01:10 Then we have to go through each of the other ones as well, assuming that you don't roll a seven or an 11, and we'll go through each one.

01:18 So for the probability of ruling a four, right, which would be your point number, and then another four and another four would be three -36th, right? because that's my probability of ruling a four.

01:38 How do i roll a four? i can go a two in a two, two, three to one, or a one in a three times.

01:46 Now we got to figure out what would be the probability of rolling another four.

01:51 There's a couple of ways that you can figure this out.

01:56 You can think of it from what is the probability that i wouldn't roll a four or a seven.

02:03 So my probability that i wouldn't roll a four and a seven would be 27 out of 36.

02:11 So if i rolled anything else, it would be a 27 out of 36.

02:18 So which basically is just one minus, right, my 336 plus my 636.

02:30 And then you could plug it into an equation and the equation is a1 all over 1 minus r...

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