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Chapter 3: Problem 18

If a fair coin is tossed at random five independent times, find the conditional probability of five heads given that there are at least four heads.

Short Answer

Expert verified
The conditional probability of getting 5 heads given that there are at least 4 heads is 0.1667.

Step by step solution

01

Calculate the number of ways to get 5 heads

When flipping a coin 5 times, there's only 1 way to get 5 heads 鈥� that is, when all 5 outcomes are heads. Hence, \( P(5 \text { heads }) = \binom{5}{5}(0.5)^5(0.5)^0 = 1 \times 0.03125 = 0.03125\) where \(\binom{n}{k}\) is the binomial coefficient and represents the number of ways to choose k successes (heads in this case) out of n trials.
02

Calculate the number of ways to get at least 4 heads

This can occur in two ways: either getting 4 heads and 1 tail or getting 5 heads. Let's calculate these two probabilities separately. For 4 heads and 1 tail, the probability is given by \( P(4 \text { heads }) = \binom{5}{4}(0.5)^4(0.5)^1 = 5 \times 0.03125 = 0.15625\) . Adding this to the probability of getting 5 heads, we get the total probability of getting at least 4 heads, which is \(0.15625 + 0.03125 = 0.1875\).
03

Calculate the conditional probability

The conditional probability of A given B, denoted \( P(A|B) \), is defined as \( P(A \cap B) / P(B) \). In this case, we want to calculate the conditional probability of getting 5 heads given that we've gotten at least 4 heads. This is simply the ratio of the probability of getting 5 heads to the probability of getting at least 4 heads, so \( P(5 \text{ heads } | \text{ at least 4 heads }) = P(5 \text{ heads }) / P(\text{ at least 4 heads }) = 0.03125 / 0.1875 = 0.1667\).

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