Probability
Probability of drawing until 3 consecutive A's
MJAT_TS4_P1
Grade 12
Question:
A person has 6 cards: A, K, 2, Q, 10, 9. The person randomly draws the cards one by one with replacement till he gets 3 consecutive A's. If $P_n$ represents the probability that at least $n$ cards are drawn, then:
A) $216P_5 - 180P_4 = 5P_2 + 30P_3$
B) $180P_5 - 216P_4 = 30P_2 + 5P_3$
C) $216P_5 + 180P_4 = 5P_2 - 30P_3$
D) $180P_5 + 216P_4 = 30P_2 - 5P_3$
Step-by-Step Solution
Key Concept: $P(\text{draw A}) = 1/6$. $P_n = P(\text{at least }n\text{ draws needed})$. $P_1=1$. Establish a recurrence using conditional probabilities based on how many consecutive A's appear at the end of the first $n-1$ draws. The recurrence relates $P_n$ to earlier terms.
From the recurrence on $P_n$: $216P_5-180P_4=5P_2+30P_3$. Answer: **A**.
Correct Answer: A