Problem 50 - A First Course in Probability

Question

Suppose that a biased coin that lands on heads with probability \(p\) is flipped 10 times. Given that a total of 6 heads results, find the conditional probability that the first 3 outcomes are (a) \(h, t, t\) (meaning that the first flip results in heads, the second in tails, and the third in tails); (b) \(t, h, t\)

Step-by-Step Solution

Answer

(a) The conditional probability for h, t, t in the first three flips is: P(h, t, t | 6 heads) = \(\frac{p * (1-p)^2 * \binom{7}{5} * p^5 * (1-p)^2}{\binom{10}{6} * p^6 * (1-p)^4}\) (b) The conditional probability for t, h, t in the first three flips is: P(t, h, t | 6 heads) = \(\frac{(1-p) * p * (1-p) * \binom{7}{5} * p^5 * (1-p)^2}{\binom{10}{6} * p^6 * (1-p)^4}\)

Step 1: Compute binomial probability for 6 heads in 10 flips

To compute the probability of obtaining exactly 6 heads in 10 flips, we need to use the binomial probability formula: P(X=k) = \(\binom{n}{k}\) * \(p^k\) * \((1-p)^{n-k}\) In this case, X is the number of heads, n is the number of flips (10), k is the desired number of heads (6), and p is the probability of obtaining a head. Plugging in our values, we get: P(X=6) = \(\binom{10}{6}\) * \(p^6\) * \((1-p)^{4}\) Step 2: Determine the probability of the given scenarios (a) Probability for h, t, t in the first three flips

Step 2: Compute binomial probability for remaining flips and scenario (a) probability

Since the first flip has one head, we need to find the probability of obtaining 5 heads in the remaining 7 flips. We can again use the binomial probability formula: P(Y=5) = \(\binom{7}{5}\) * \(p^5\) * \((1-p)^{2}\) Now, we need to find the probability of the specific sequence h, t, t, followed by 5 heads in the remaining 7 flips: P(h, t, t | 6 heads) = \(P(h) * P(t) * P(t) * P(Y=5)\) P(h, t, t | 6 heads) = \(p * (1-p) * (1-p) * \binom{7}{5} * p^5 * (1-p)^2\) (b) Probability for t, h, t in the first three flips

Step 3: Compute binomial probability for remaining flips and scenario (b) probability

In this scenario, we have one head in the first three flips, so we need to find the probability of obtaining 5 heads in the remaining 7 flips, which is the same as in (a): P(Y=5) = \(\binom{7}{5}\) * \(p^5\) * \((1-p)^{2}\) Now, we need to find the probability of the specific sequence t, h, t, followed by 5 heads in the remaining 7 flips: P(t, h, t | 6 heads) = \(P(t) * P(h) * P(t) * P(Y=5)\) P(t, h, t | 6 heads) = \((1-p) * p * (1-p) * \binom{7}{5} * p^5 * (1-p)^2\) Step 3: Compute the conditional probabilities (a) Conditional probability for h, t, t in the first three flips

Step 4: Compute conditional probability for scenario (a)

To find the conditional probability for scenario (a), divide the probability of scenario (a) by the probability of 6 heads in 10 flips: P(h, t, t | 6 heads) = \(\frac{p * (1-p) * (1-p) * \binom{7}{5} * p^5 * (1-p)^2}{\binom{10}{6} * p^6 * (1-p)^4}\) (b) Conditional probability for t, h, t in the first three flips

Step 5: Compute conditional probability for scenario (b)

Similarly, to find the conditional probability for scenario (b), divide the probability of scenario (b) by the probability of 6 heads in 10 flips: P(t, h, t | 6 heads) = \(\frac{(1-p) * p * (1-p) * \binom{7}{5} * p^5 * (1-p)^2}{\binom{10}{6} * p^6 * (1-p)^4}\) Now you have found the conditional probabilities for each scenario.