Question

Example Consider the following dice game. A pair of standard ( fair ) dice are repeatedly...

Example

Consider the following dice game. A pair of standard ( fair ) dice are repeatedly rolled. If a ’ 7 ’ comes up before an ’ 11 ’ , then the player wins, otherwise the player loses. Let W be the event that the player wins. Find P(W). To say the dice are fair is equivalent to assuming that Laplace’s rule holds and the 36 possible outcomes for a throw of the dice are equally likely. For convenience, an addition table is presented in Table 1. Table 1: Sum of Spots on Two Dice + 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12

Solution

Let F be the event that the game ends on the first roll; let L be the event that the game ends on some later roll. Write P(W) = P(W ∩F) + P(W ∩L) = P(W ∩F) + P(W|L)P(L) Now P(W∩F) is just the probability the first roll is a seven, so P(W∩F) = 1 6. The game extends past the first roll if the player neither wins nor loses on the first roll, which is the probability the first roll is neither a seven nor an eleven. So P(L) = 1−1 6− 1 18. Now here’s the part that requires a little thought: dice have no memory, hold no grudges, so the chances you eventually win aren’t any greater or smaller if the game goes to a second roll than they were when the player first picked up the dice. Hence P(W|L) = P(W). Putting all this together, one has P(W) = 1 6 + P(W)(1− 1 6 − 1 18 ) 13 Solving for P(W) one obtains P(W) = 1 6 1 6 + 1 18 = 3 4

******please do the following exercise *******

Exercise 6.

. Generalize the last example. Let A and B be two disjoint events, e.g. rolling a seven and rolling an even number, respectively. If A < B is the event that A happens before B in repeated rolls,

1. show that P(A < B) = P(A) P(A) + P(B)

2. If 2A < B denotes the event at A happens twice before B happens, find an expression for P(2A < B)

. 3. Find the probability that in repeated rolls of a pair of fair dice one rolls an eleven before two rolls of a seven occur.

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Answer 6)

Given the events: A=rolling a 7 B = rolling an even number A<B rolling a 7 before an eve number Now, P(An(AU B)) P(A < B) = P(A l (AUB)) = Now, we know that: P(An (AU B))P(A) and, because A and B are disjoint events, P(AUB) P(A) P(B) Hence, PlA < B) = P(A | (AU B)) = P(A) + P(B) P(A)2) Now, clearly as above, and as rolling 7 twice and rolling an even number are disjoint events, Hence, P(A1 n A2) P(A) P(A2) P(A) P(A) P(A) P(2A < B)- P(Ai n A2) + P(B) P(2A < B) = P(A) * P(A)] + P(B) Pl2A < B)-P( アイア(B)3) P(11) P(11u (7n7)) P(11 < 2(7))-P(11 | (11 U (7n7))) P(112(7 3636

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