Question

# A card is drawn from a deck of 52 cards. Find the probability of getting a king or a heart or a red card.

Solution

## Let S be the sample space. Then, n(S) = 52. Let E1,E2 and E3 be the events of getting a king, a heart and a red card respectivel. Then, n(E1)=4,n(E2)=13 and n(E3)=26. (E1∩E2) = event of getting a king of hearts; (E2∩E3) = event of getting a heart            [∵ a heart is a red card also]; (E3∩E1) = event of getting a red king; and (E1∩E2∩E3) = event of getting a king of hearts. ∴n(E1∩E2)=1,n(E2∩E3)=13,n(E3∩E1)=2 and n(E1∩E2∩E3)=1. ∴P(E1)=n(E1)n(S)=452=113;P(E2)=n(E2)n(S)=1352=14; P(E3)=n(E3)n(S)=2652=12;P(E1∩E2)=n(E1∩E2)n(S)=152; P(E2∩E3)=n(E2∩E3)n(S)=252=126 and P(E1∩E2∩E3)=n(E1∩E2∩E3)n(S)=152. ∴ P(getting a king or a heart or a red card) ∴ P(getting a king or a heart or a red card) =P(E1 or E2 or E3)=P(E1∪E2∪E3) =P(E1)+P(E2)+P(E3)−P(E1∩E2)−P(E2∩E3)−P(E3∩E1)+P(E1∩E2∩E3) =(113+14+12−152−14−126+152)=2852=713. Hence, the required probability is 713. Mathematics

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