Suppose that a biased coin that lands on heads with...

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) $t, h, t$

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) $t, h, t$

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Key Concepts

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has occurred. In problems involving coin flips, this concept allows us to adjust our calculations based on confirmed outcomes, such as knowing the total number of heads, and then determining the probability of a more specific sequence of results given that information.

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials with two outcomes, such as heads or tails in coin flips. This distribution is fundamental to problems where each flip is a Bernoulli trial with a specific probability of success, and it often underpins calculations when we condition on a certain count of successes.

Combinatorial Analysis

Combinatorial analysis involves counting the number of ways events can occur, often using binomial coefficients to count arrangements in sequences. In the context of coin flipping under certain conditions, combinatorial methods are used to determine how many sequences meet specific criteria, such as having a fixed total number of heads or matching a particular pattern in predetermined positions.

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