Question

a fair coin is flipped three times. events a and b are defined as: a: thwere are at least two consecutive heads somewhere in the sequence b: at least one flipc omes up tails what is p(a|b)?

Answer 1

The probability of event A occurring given that event B has already occurred (P(A|B)) is 1/4.

Step-by-step explanation:

In this question, we are given the events A and B.

Event A is defined as having at least two consecutive heads somewhere in the sequence.

Event B is defined as having at least one flip come up tails.

We need to find the conditional probability P(A|B), which is the probability of event A occurring given that event B has already occurred.

To find P(A|B), we can use the formula:

P(A|B) = P(A ∩ B) / P(B).

Let's break it down step by step:

1. Event A can occur in several ways: HHT, HTH, and THH. So, P(A) = 3/8 (since there are 8 possible outcomes when flipping a coin 3 times).
2. Event B can occur in several ways: TTT, TTH, THT, HTT, but it cannot occur when the result is all heads (HHH). So, P(B) = 4/8 = 1/2 (since there are 8 possible outcomes when flipping a coin 3 times).
3. The intersection of events A and B (A ∩ B) occurs only when we get the specific outcome THH. So, P(A ∩ B) = 1/8 (since there are 8 possible outcomes when flipping a coin 3 times).
4. Now, we can calculate P(A|B) using the formula: P(A|B) = P(A ∩ B) / P(B) = (1/8) / (1/2) = 1/4.

Therefore, the probability P(A|B) is 1/4.