Probability
Game Probability (Geometric Series)
nta_pyq_2025_apr
Grade 12
Question:
$A$ and $B$ alternately throw a pair of dice. $A$ wins if he throws a sum of 5 before $B$ throws a sum of 8, and $B$ wins if he throws a sum of 8 before $A$ throws a sum of 5. The probability that $A$ wins if $A$ makes the first throw, is:
$\frac{8}{17}$
$\frac{9}{19}$
$\frac{9}{17}$
$\frac{8}{19}$
Step-by-Step Solution
Key Concept: Find $P(A \text{ scores } 5) = p$ and $P(B \text{ scores } 8) = q$ from standard dice. Then $P(A \text{ wins}) = \frac{p}{p + q - pq}$ (geometric series sum).
$P(\text{sum}=5) = 4/36$, $P(\text{sum}=8) = 5/36$. $P(\bar{A}) = 32/36$, $P(\bar{B}) = 31/36$. $P(A \text{ wins}) = \frac{4/36}{1 - (32/36)(31/36)} = \frac{4/36}{(36^2 - 32\times31)/36^2} = \frac{4 \times 36}{36^2 - 992} = \frac{144}{304} = \frac{9}{19}$.
Correct Answer: $\frac{9}{19}$