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Each of the persons A and B independently tosses three fair coins. The probability that both of them get the same number of heads is :

JEE · Math · previous-year question

  1. A.$${1 \over 8}$$
  2. B.$${5 \over 8}$$
  3. C.$${5 \over 16}$$correct
  4. D.1

Answer

C. $${5 \over 16}$$

Explanation

Let x be the number of heads obtained by A, and y be the number of heads obtained by B. Note that x and y are binomial variable with parameters n = 3 and p = $${1 \over 2}$$ $$\therefore$$ Probability that both A and B obtained the same number of heads is $$ = P(x = 0)\,.\,P(y = 0) + P(x = 1)\,.\,P(y = 1) + P(x = 2)\,.\,P(y = 2) + P(x = 3)\,.\,P(y = 3)$$ $$ = {\left[ {{3_{{C_0}}}{{\left( {{1 \over 2}} \right)}^3}} \right]^2} + {\left[ {{3_{{C_1}}}{{\left( {{1 \over 2}} \right)}^3}} \right]^2} + {\left[ {{3_{{C_2}}}{{\left( {{1 \over 2}} \right)}^3}} \right]^2} + {\left[ {{3_{{C_3}}}{{\left( {{1 \over 2}} \right)}^3}} \right]^2}$$ $$ = {\left( {{1 \over 2}} \right)^6}\left[ {1 + 9 + 9 + 1} \right]$$ $$ = {{20} \over {64}}$$ $$ = {5 \over {16}}$$

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