Final answer:
The probability that the first three or last three flips of a fair coin come up heads is 11/32. This is calculated by adding the probabilities of each scenario and subtracting the overlap.
Step-by-step explanation:
The subject of this question is probability, which is a branch of Mathematics.
To find the probability that the first three flips come up heads or the last three flips come up heads when a fair coin is flipped 5 times, we consider the two scenarios separately and then apply the addition rule of probability.
First three flips are heads (HHHXX):
The probability for each head (H) is 0.5, and since each flip is independent, we multiply the probabilities: 0.5 * 0.5 * 0.5 = 0.125 (or 1/8). The last two flips (XX) can be any combination, so we don't need to consider them for this part of the question.
Last three flips are heads (XXHHH):
By similar reasoning, the probability is also 0.5 * 0.5 * 0.5 = 0.125 (or 1/8).
However, we need to account for the overlap where the first three and last three flips are all heads (HHHHH), which we've so far included in both scenarios. This overlapping case has a probability of 0.5^5 = 0.03125 (or 1/32). We subtract this from the sum of the individual probabilities to avoid double-counting.
The total probability is (1/8 + 1/8 - 1/32) which equals 2/8 - 1/32 = 1/4 - 1/32 = 8/32 - 1/32 = 7/32.
Therefore, the correct answer is D) 11/32.
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