Final answer:
To find the probability of getting the fifth tail on the ninth coin flip, we calculate the number of ways to get 4 tails in the first 8 flips, then multiply by the probability of this happening, and finally, consider the 9th flip must also be a tail.
Step-by-step explanation:
To find the probability that a person flipping a coin gets the fifth tail on the ninth flip, we are dealing with a binomial probability problem. We need exactly 4 tails in the first 8 flips and the 9th flip must be a tail. Let's denote 'T' as a Tail and 'H' as a Head. The flips can be represented as TTTTHHHH (not in order), followed by an additional T on the 9th flip.
First, we calculate the number of ways to arrange 4 tails in 8 flips. This can be done using the combination formula: C(n, k) = n! / (k! * (n-k)!), where n is the total number of flips, k is the number of tails, and ! represents factorial. Here, n=8 and k=4, so the number of combinations is C(8, 4).
Then we calculate the probability of having exactly 4 tails and 4 heads in these combinations, which is (0.5)^4 * (0.5)^4. Finally, we need to consider the 9th flip, which must be a tail with a probability of 0.5.
Therefore, the final probability is C(8, 4) * (0.5)^4 * (0.5)^4 * 0.5.