A fair coin is tossed 5 times.
What is the probability that exactly 3 heads are obtained?Solution
The coin is tossed 5 times, hence the number of trials is n=5.
The coin being a fair one, the outcome of a head in one toss has a probability p=0.5 and an outcome of a tail in one toss has a probability 1−p=0.5
The probability of having 3 heads in 5 trials is given by the formula for binomial probabilities above with n=5, k=3 and p=0.5
P(3heads in 5 trials)=(53)(0.5)3(1−0.5)5−3=(53)(0.5)3(0.5)2
Use formula for combinations to calculate
(53)=5!3!(5−3)!=1×2×3×4×5(1×2×3)(1×2)=10
Substitute
P(3heads in 5 trials)=10(0.5)3(0.5)2=0.3125
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