Chapter 3: Q. 3.74 (page 104)

$A$ and $B$ alternate rolling a pair of dice, stopping either when $A$ rolls the sum $9$ or when $B$ rolls the sum $6$ . Assuming that $A$ rolls first, find the probability that the final roll is made by $A$ .

Short Answer

Expert verified

Probability that the final roll made by $A$ is $\frac{9}{19}$ where assuming that $A$ rolls first

Step by step solution

Find the probability of A rolls then B rolls

A- $A$ rolls $9$ times in a certain order.

B- $B$ rolls $6$ dice in a certain order.

Probabilities:

$P (A) = \frac{4}{36} = \frac{1}{9}, A = {(3, 6), (4, 5), (5, 4), (6, 3)}$

$P (B) = \frac{5}{36}, B = {(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)}$

$A$ and $B$ can only appear in separate rolls and are thus distinct.

The likelihood that neither $A$ nor $B$ will roll their winning number in one turn $(A a n d B)$ is

$P [(A \cup B)^{c}] = 1 - P (A \cup B)$

$= 1 - [P (A) - P (B) - P (A B)]$

$= 1 - [P (A) - P (B) - P (A) P (B)]$

$= 1 - [\frac{4}{36} + \frac{5}{36} - \frac{4}{36} \times \frac{5}{36}]$

$= \frac{62}{81}$

Find the Probability that the final roll made by A

The likelihood of $A$ finishing the game is the sum of the probabilities of mutually exclusive situations in which $A$ rolls $9$ after $i = 0, 1, 2, 3, . . .$ turns in which neither player wins. Each turn's outcome is distinct.

$P$ [" $A$ ends the game"] localid="1649572019674" $= \sum_{i \in ℕ} [P {[(A \cup B)^{c}]}^{i} P (A)$

$= \sum_{i \in ℕ} {(\frac{62}{81})}^{i} \frac{1}{9}$

$= \frac{1}{9} \sum_{i \in ℕ} {(\frac{62}{81})}^{i}$

$= \frac{1}{9} \frac{1}{1 - \frac{62}{81}}$

$= \frac{9}{19}$

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$A$ and $B$ alternate rolling a pair of dice, stopping either when $A$ rolls the sum $9$ or when $B$ rolls the sum $6$ . Assuming that $A$ rolls first, find the probability that the final roll is made by $A$ .

Short Answer

Step by step solution

Find the probability of A rolls then B rolls

Find the Probability that the final roll made by A

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Aand Balternate rolling a pair of dice, stopping either when Arolls the sum 9or when Brolls the sum 6. Assuming that A rolls first, find the probability that the final roll is made by A.

Short Answer

Step by step solution

Find the probability of A rolls then B rolls

Find the Probability that the final roll made by A

One App. One Place for Learning.

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$A$ and $B$ alternate rolling a pair of dice, stopping either when $A$ rolls the sum $9$ or when $B$ rolls the sum $6$ . Assuming that $A$ rolls first, find the probability that the final roll is made by $A$ .