The casino game of craps is played as follows: Step 1: The player rolls a pair of standard 6-sided dice and takes their sum a. If the sum is 7 or 11, then the player wins and the game is over b. If the sum is 2, 3, or 12, then the player loses (this is called “crapping out”) and the game is over c. If the sum is anything else, then we record the sum (lets call it “X”) and continue to the next step Step 2: The player then rerolls the dice and takes their sum a. If the sum is X, the player wins and the game is over b. If the sum is 7, the player loses and the game is over c. If the sum is anything else, repeat step 2. Now suppose that you notice something odd - one of the two dice isn’t balanced that well, and always comes up in the range 2-5 (with equal probability) but never 1 or 6. 1. For each number between 2 and 12, what is the probability of rolling the dice so that they sum to that number? 2. a. What’s the probability of winning on the very first roll? b. What’s the probability of losing (“crapping out”) on the very first roll? 3. Suppose that on the first roll, you do not win or lose, but rather, you get the sum X, which has roll probability p. Given that you have already made it to this point, what’s your chance of winning going forward? 4. If you play the game of craps with these two dice, you will get one dollar if you win, and lose one dollar if you lose, then what is the expected return for playing the game?
The casino game of craps is played as follows:
Step 1: The player rolls a pair of standard 6-sided dice and takes their sum
a. If the sum is 7 or 11, then the player wins and the game is over
b. If the sum is 2, 3, or 12, then the player loses (this is called “crapping out”) and
the game is over
c. If the sum is anything else, then we record the sum (lets call it “X”) and continue
to the next step
Step 2: The player then rerolls the dice and takes their sum
a. If the sum is X, the player wins and the game is over
b. If the sum is 7, the player loses and the game is over
c. If the sum is anything else, repeat step 2.
Now suppose that you notice something odd - one of the two dice isn’t balanced that well, and always comes up in the range 2-5 (with equal
1. For each number between 2 and 12, what is the probability of rolling the dice so that
they sum to that number?
2.
a. What’s the probability of winning on the very first roll?
b. What’s the probability of losing (“crapping out”) on the very first roll?
3. Suppose that on the first roll, you do not win or lose, but rather, you get the sum X, which
has roll probability p. Given that you have already made it to this point, what’s your
chance of winning going forward?
4. If you play the game of craps with these two dice, you will get one dollar if you win, and
lose one dollar if you lose, then what is the expected return for playing the game?
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