A fair (unbiased) coin is tossed 15 times. The probability of getting exactly 8 Heads (round off to three decimal places), is ________.Correct answer is '0.196'. Can you explain this answer?

Question

Answer

Problem:
A fair (unbiased) coin is tossed 15 times. The probability of getting exactly 8 Heads (round off to three decimal places) is ________.

Solution:

To find the probability of getting exactly 8 Heads when a fair coin is tossed 15 times, we need to use the binomial probability formula. The binomial probability formula is given by:

P(x) = C(n, x) * p^x * q^(n-x)

Where:
P(x) = Probability of getting exactly x successes
C(n, x) = Combination of n items taken x at a time
p = Probability of getting a success in a single trial
q = Probability of getting a failure in a single trial
n = Total number of trials

In this case, the probability of getting a Head (success) in a single trial is 0.5, as the coin is fair. Therefore, p = 0.5. Similarly, the probability of getting a Tail (failure) in a single trial is also 0.5, so q = 0.5. The total number of trials is 15, so n = 15.

Now, we can substitute these values into the formula:

P(8) = C(15, 8) * 0.5^8 * 0.5^(15-8)

Calculating the Combination:
C(15, 8) = 15! / (8! * (15-8)!) = 3003

Calculating the Probability:
P(8) = 3003 * 0.5^8 * 0.5^7 = 3003 * 0.5^15

Rounding off the Probability:
P(8) = 3003 * 0.000030517578125 ≈ 0.091

Therefore, the probability of getting exactly 8 Heads when a fair coin is tossed 15 times is approximately 0.091.

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