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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 five heads given that there are at least four heads is 0.16667.

Step by step solution

01

Calculate P(H_{5})

First calculate the probability of getting five heads in five tosses. Plug \(n=5\), \(x=5\), and \(p=0.5\) into the binomial formula to find \(P(H_{5}) = C(5, 5)(0.5)^5(1-0.5)^{5-5} = 0.03125\).
02

Calculate P(H_{4+})

Next, calculate the probability of getting at least four heads in five tosses. This could happen in two ways: either exactly four heads or exactly five heads, so we calculate both probabilities and add them together. We already know \(P(H_{5})=0.03125\). Plug \(n=5\), \(x=4\), and \(p=0.5\) into the binomial formula to find \(P(H_{4}) = C(5, 4)(0.5)^4(1-0.5)^{5-4} = 0.15625\). Add these probabilities to find \(P(H_{4+}) = P(H_{4}) + P(H_{5}) = 0.15625 + 0.03125 = 0.1875\).
03

Calculate Conditional Probability

Finally, divide \(P(H_{5})\) by \(P(H_{4+})\) to find the conditional probability of getting five heads given at least four heads: \(P(H_{5}|H_{4+}) = \frac{P(H_{5})}{P(H_{4+})} = \frac{0.03125}{0.1875} = 0.16667.\)

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