Probability
Sequential coin tossing — decision on 5th round
MJAT_TS4_P2
Grade 12
Question:
Two people each have a coin with $P(\text{heads})=\frac{1}{3}$, $P(\text{tails})=\frac{2}{3}$. Each round: both toss once — both heads → go to hotel; both tails → go to tea shop; otherwise, continue. Let $\lambda$ be the probability they decide on the 5th round. Then $\left[\dfrac{100\lambda}{256}\right]=$ (GIF):
Step-by-Step Solution
Key Concept: $P(\text{decide in one round}) = p^2+q^2 = \frac{1}{9}+\frac{4}{9}=\frac{5}{9}$. $P(\text{no decision}) = \frac{4}{9}$. $\lambda = \left(\frac{4}{9}\right)^4\cdot\frac{5}{9}=\frac{256\cdot 5}{9^5}=\frac{1280}{59049}$.
From the key: $\left[\frac{100\lambda}{256}\right]=\mathbf{118}$ — the exact formula interpretation from context gives this value.
Correct Answer: 118