I flip a fair coin seven times in a row. What is the probability that I get at most five
heads?
To calculate the probability of getting at most five heads when flipping a fair coin seven times in a row, we need to consider the different scenarios that satisfy this condition.
Let's break it down into three parts:
1. Probability of getting exactly zero heads: This occurs when we get seven tails in a row. The probability of getting a tail on a single coin flip is 1/2, so the probability of getting seven tails in a row is (1/2)^7 = 1/128.
2. Probability of getting exactly one head: This can happen in seven different ways (e.g., H-T-T-T-T-T-T, T-H-T-T-T-T-T, etc.). The probability of getting a head on a single coin flip is also 1/2, so the probability of getting exactly one head is 7*(1/2)^7 = 7/128.
3. Probability of getting exactly two, three, four, or five heads: Similarly, these scenarios can occur in multiple ways and their probabilities can be calculated using combinations. The probability of getting exactly two heads is (7 choose 2) * (1/2)^7 = 21/128, the probability of getting exactly three heads is (7 choose 3) * (1/2)^7 = 35/128, the probability of getting exactly four heads is (7 choose 4) * (1/2)^7 = 35/128, and the probability of getting exactly five heads is (7 choose 5) * (1/2)^7 = 21/128.
To find the overall probability of getting at most five heads, we need to sum up the probabilities from each part:
P(at most five heads) = P(zero heads) + P(one head) + P(two heads) + P(three heads) + P(four heads) + P(five heads)
= 1/128 + 7/128 + 21/128 + 35/128 + 35/128 + 21/128
= 120/128
= 15/16
Therefore, the probability of getting at most five heads when flipping a fair coin seven times in a row is 15/16.