energetics wrote:
If a fair two-sided coin is flipped 6 times, what is the probability that tails is the result at least twice but at most 5 times?
A) \(\frac{5}{8}\)
B) \(\frac{3}{4}\)
C) \(\frac{7}{8}\)
D) \(\frac{57}{64}\)
E) \(\frac{15}{16}\)
We need to determine the following probabilities:
TTHHHH, TTTHHH, TTTTHH, and TTTTTH
P(TTHHHH) = (1/2)^6 = 1/64
TTHHHH can be arranged in 6C2 = 6!(2! x 4!) = (6x5)/2 = 15 ways, so the probability is 15/64.
P(TTTHHH) = (1/2)^6 = 1/64
TTTHHH can be arranged in 6C3 = 6!(3! x 3!) = (6x5x4)/3! = 20 ways, so the probability is 20/64.
P(TTTTHH) = (1/2)^6 = 1/64
TTTTHH can be arranged in 6C4 = 6!(2! x 4!) = (6x5)/2 = 15 ways, so the probability is 15/64.
P(TTTTTH) = (1/2)^6 = 1/64
TTTTTH can be arranged in 6C5 = 6!/5! = 6 ways, so the probability is 6/64.
Thus, the total probability is (15 + 20 + 15 + 6)/64 = 56/64 = 7/8.
Answer: C
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