Multiple choice

A fair coin is tossed at a fixed number of times. If the probability of getting exactly $3$ heads equals the probability of getting exactly $5$ heads, then the probability of getting exactly one head is

  1. $\dfrac{1}{64}$
  2. $\dfrac{1}{32}$
  3. $\dfrac{1}{16}$
  4. $\dfrac{1}{8}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Binomial probability P(X=k) = nCk * (1/2)^n. P(X=3) = P(X=5) implies nC3 = nC5. This means 3+5 = n, so n=8. Probability of one head = 8C1 * (1/2)^8 = 8 / 256 = 1/32.

AI explanation

Using the binomial distribution formula, the probability of k heads in n tosses is (n choose k) times (1/2) to the power of n. Since the probability of exactly 3 heads equals the probability of exactly 5 heads, (n choose 3) equals (n choose 5), which implies n = 3 + 5 = 8 total tosses. The probability of getting exactly one head is (8 choose 1) times (1/2) to the power of 8, which equals 8/256 or 1/32.

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