What is the probability of obtaining exactly five heads in six flips of a coin, given that at least one is a head?
To calculate the probability of obtaining exactly five heads in six flips of a coin, given that at least one is a head, we need to consider two scenarios:
Scenario 1: Exactly one head and four tails in the first five flips, followed by a head in the last flip.
Scenario 2: Exactly five heads and one tail in the first five flips, followed by a head in the last flip.
We can calculate the probability of each scenario and then add them together to get the total probability.
Scenario 1:
The probability of getting a head in a single flip of a fair coin is 1/2, and the probability of getting tails is also 1/2.
So, the probability of getting exactly one head and four tails in five flips is (1/2)^5.
Scenario 2:
The probability of getting five heads and one tail in six flips is (1/2)^6, since each flip is independent and has a 1/2 chance of being a head or a tail.
Now, let's calculate the probabilities of each scenario separately:
Scenario 1: (1/2)^5 = 1/32
Scenario 2: (1/2)^6 = 1/64
Since either Scenario 1 or Scenario 2 must occur to meet the condition of at least one head, we can add the probabilities together:
Probability of obtaining exactly five heads in six flips, given that at least one is a head = Scenario 1 + Scenario 2
= 1/32 + 1/64
= 3/64
Therefore, the probability of obtaining exactly five heads in six flips of a coin, given that at least one is a head, is 3/64.