Question

Two cards are selected at random without replacement from a well-shuffled deck of 52 playing cards. Find the probability of the given event. (Round your answer to three decimal places.) A pair is not drawn.

Answer 1

The probability that a pair is not drawn  = 0.765

Step-by-step explanation:

Here, the total number of cards in the given deck = 52

The total number of suits = 4 (13 cards each)

Now, P( Drawing 2 clubs) = $\frac{^{13}C_2}{^{52}C_2} = \frac{78}{1326} = (\frac{1}{17} )$

Similarly,  P( Drawing 2 diamond ) = $\frac{^{13}C_2}{^{52}C_2} = \frac{78}{1326} = (\frac{1}{17} )$

P( Drawing 2 spades) = $\frac{^{13}C_2}{^{52}C_2} = \frac{78}{1326} = (\frac{1}{17} )$

P( Drawing 2 hearts ) = $\frac{^{13}C_2}{^{52}C_2} = \frac{78}{1326} = (\frac{1}{17} )$

⇒ Probability of drawing 2 clubs or 2 spades or 2 hearts or 2 diamonds

=  P( Drawing 2 clubs) +  P( Drawing 2 diamond ) P( Drawing 2 spades)+ P( Drawing 2 hearts )  = $(\frac{1}{17})+ (\frac{1}{17})+ (\frac{1}{17})+ (\frac{1}{17}) = 4\times (\frac{1}{17}) = \frac{4}{17} = 0.2352$

Now, P(NOT DRAWING A PAIR) = 1 - P(DRAWING a PAIR)

                                                       = 1- 0.2352 = 0.7647

Hence, the probability that a pair is not drawn  = 0.7647