Multiple choice

Two players A and B each toss $10$ coins. The probability that they show equal number of heads is?

  1. $\dfrac{^{20}C_{10}}{2^{20}}$
  2. $\dfrac{1}{2^{10}}$
  3. $\dfrac{1}{2^{20}}$
  4. $\dfrac{1}{2^9}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Total outcomes = 2^10 * 2^10 = 2^20. Favorable outcomes: A gets k heads, B gets k heads. Sum of (10Ck)^2 for k=0 to 10 = 20C10. Probability = 20C10 / 2^20.

AI explanation

If both players show an equal number of heads, player B must get exactly k heads whenever player A gets k heads. Using the binomial probability formula, the probability of both getting exactly k heads is the product of their individual probabilities, which is (10Ck / 2^10) * (10Ck / 2^10). Applying Vandermonde's identity, the summation of (10Ck)^2 from k=0 to 10 equals 20C10. Therefore, the total probability of both showing an equal number of heads is 20C10 / 2^20.

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