Question

A card is drawn from a standard deck of cards and then replaced find the following probabilities P(spade and spade)P(queen and ace)P(face card and face card)P(king and king and king)

Answer 1

1) P(spade and spade)

First of all, a deck has 52 cards, 13 spades.

The probability that we want to calculate is:

P(spade and spade) = P(spade_1) * P(spade_2)

We calculate the probability of drawn a spade:

P(spade_1) = probability of drawn a spade = # of spades / # of cards in the deck = 13/52 = 1/4

After with drawn the card, we insert the card again in the deck, so the probability of drawn a second spade is again:

P(spade_2) = 13/52 = 1/4

Because again we have the same number of spades and cards in the deck.

P(spade and spade) = P(spade_1) * P(spade_2) = (1/4) * (1/4) = 1/16 = 0.0625 or 6.25%

NOTE: The key with these problems is that we are raising a card and then putting the card again in the deck.

2) P(queen and ace)

P(queen) = # of queens / # of cards in the deck = 4/52 = 1/13

P(ace) = # of aces / # of cards in the deck = 4/52 = 1/13

P(queen and ace) = P(queen) * P(ace) = (1/13) * (1/13) = 1/169 = 0.0059 or 0.59 %

3) P(face card and face card)

P(face card) = # of face cards / # of cards in the deck = (4*3)/52 = 12/52 = 0.23

P(face card and face card) = P(face card) * P(face card) = (12/52) * (12/52) = 9/169 = 0.053 or 5.32%

4) P(king and king and king)

P(king) = # of kings / # of cards in the deck = 4/52 = 1/13

P(king and king and king) = P(king) * P(king) * P(king) = (1/13) * (1/13) * (1/13) = 1/2197 = 0.000455 or 0.0455%