Multiple choice

$A$ and $B$ toss 3 coins. The probability that both obtain same number of tails is $p$ and the probability that both obtain same number of heads is $q$, then the value of $p+q$ is

  1. $\dfrac{5}{16}$
  2. $\dfrac{10}{16}$
  3. $\dfrac{12}{16}$
  4. $\dfrac{7}{16}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For 3 coins, the total outcomes are 2^3 = 8. The probability of getting k tails is C(3,k)/8. Let T be number of tails and H be number of heads. P(same tails) = sum(P(A=k) * P(B=k)) = sum((C(3,k)/8)^2) = (1/64 + 9/64 + 9/64 + 1/64) = 20/64 = 5/16. Similarly, P(same heads) = 5/16. Total p+q = 10/16.

AI explanation

When three coins are tossed, the probabilities of obtaining zero, one, two, and three tails are 1/8, 3/8, 3/8, and 1/8 respectively. For both individuals to obtain the same number of tails, we sum the squared probabilities for each matching outcome. This gives 1/8 squared plus 3/8 squared plus 3/8 squared plus 1/8 squared, which is 1/64 plus 9/64 plus 9/64 plus 1/64, totaling 20/64 or 5/16. Since the number of heads is also perfectly symmetric, the probability that both obtain the same number of heads is the same, making p plus q equal to 5/16 plus 5/16, which is 10/16.

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