What is the probability of obtaining exactly four heads in five flips of a coin, given that at least one is a head?
1
0
To calculate the probability of obtaining exactly four heads in five flips of a coin, given that at least one is a head, we can use the concept of conditional probability.
First, let's determine the total number of possible outcomes when flipping a coin five times. Since each flip can result in either heads or tails, there are a total of 2^5 = 32 possible outcomes.
Next, let's find the number of outcomes where at least one flip results in heads. To calculate this, we subtract the number of outcomes where all flips result in tails from the total number of outcomes. The number of outcomes where all flips result in tails is simply 1 (since there is only one way to get all tails in five flips). Therefore, the number of outcomes where at least one flip results in heads is 32 - 1 = 31.
Now, let's calculate the number of outcomes where exactly four flips result in heads. To do this, we need to consider the different ways we can choose four flips out of the five to be heads. This can be calculated using the combination formula: C(n, k) = n! / (k!(n-k)!), where n is the total number of flips (5 in this case) and k is the number of flips that result in heads (4 in this case). Plugging in the values, we have C(5, 4) = 5! / (4!(5-4)!) = 5.
Finally, we can calculate the probability by dividing the number of outcomes where exactly four flips result in heads (5) by the number of outcomes where at least one flip results in heads (31). Therefore, the probability is 5/31 ≈ 0.161.
So, the probability of obtaining exactly four heads in five flips of a coin, given that at least one is a head, is approximately 0.161 or 16.1%.