# Probability Problem with playing cards

A standard pack of 52 playing cards is shuﬄed thoroughly and then cut. The pack is then shuﬄed and cut for a second time. Within the pack, a 'picture' card is deﬁned to be a card showing an ace, king, queen or jack (that is, not a card showing any of the numbers 2, 3,... , 10).

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Solution involves the use of basic probability formulae, features of a deck of playing cards.

Permutations/combinations with and without repetitions are used.

In general, you should give your answers as fractions. Where you give an answer as a decimal, you should give it correct to four decimal places.

(a) In this part of the question, state which rule of probability you use for each calculation.

A standard pack of 52 playing cards is shuffled thoroughly and then cut. The pack is then shuffled and cut for a second time. Within the pack, a 'picture' card is deﬁned to be a card showing an ace, king, queen or jack (that is, not a card showing any of the numbers 2, 3,... , 10).

(i) Find the probability that the ﬁrst time the pack is cut, the card displayed is a picture card.

Solution: There are four suits (clubs (♣), diamonds (♦), hearts (♥) and spades (♠)) in any standard pack of 52 cards. Each suit has 13 cards in it. Each suit contains four picture cards. Hence there are 16 picture cards in a standard pack of 52 playing cards. Remaining ...

#### Solution Summary

This solution explains how to solve a few problems on probability based on the features of a set of playing cards.