Multiple choice

From a well shuffled pack of $52$ playing cards two cards drawn at random. The probability that either both are red or both are kings is:

  1. $\dfrac{{\left( {^{26}{C_2} + \,{\,^4}{C_2}} \right)}}{{^{52}{C_2}}}$
  2. $\dfrac{{\left( {^{26}{C_2} + {\,^4}{C_2} - {\,^2}{C_2}} \right)}}{{^{52}{C_2}}}$
  3. $\dfrac{{^{30}{C_2}}}{{^{52}{C_2}}}$
  4. $\dfrac{{^{39}{C_2}}}{{^{52}{C_2}}}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

P(A or B) = P(A) + P(B) - P(A and B). A = both red (26C2), B = both kings (4C2). A and B = both red kings (2C2). Total ways = 52C2. Probability = (26C2 + 4C2 - 2C2) / 52C2.

AI explanation

Using the addition theorem of probability, the probability of two non-mutually exclusive events A and B is P(A) + P(B) - P(A intersection B). The total ways to draw two cards is 52C2, the ways to draw two red cards is 26C2, and the ways to draw two kings is 4C2. The overlapping case of drawing two red kings gives 2C2 ways. Therefore, the required probability is (26C2 + 4C2 - 2C2) / 52C2.

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