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Chapter 2: Problem 22

If a fair coin is successively flipped, find the probability that a head first appears on the fifth trial.

Short Answer

Expert verified
The probability that a head first appears on the fifth trial is \( \frac{1}{32} \).

Step by step solution

01

Determine the probability of each flip outcome

Since the coin is fair, the probability of getting a head (H) or a tail (T) on any single flip is equal: - P(H) = 1/2 - P(T) = 1/2
02

Determine the specific sequence of outcomes for the problem

We are looking for the probability of getting a head first appears on the fifth trial, so the sequence of outcomes we are considering is: TTTT H
03

Calculate the probability of the specific sequence

Since the coin flips are independent events, we can find the probability of this specific sequence by multiplying the probabilities of each outcome in the sequence: P(TTTT H) = P(T) * P(T) * P(T) * P(T) * P(H)
04

Plug in the probabilities for each outcome

Substitute the probability for each outcome found in Step 1: P(TTTT H) = (1/2) * (1/2) * (1/2) * (1/2) * (1/2)
05

Calculate the final probability

Multiply the probabilities together: P(TTTT H) = \( \frac{1}{32} \) So, the probability that a head first appears on the fifth trial is 1/32.

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