19 A normal deck of cards has 52 cards consisting of 13 each

19. A normal deck of cards has 52 cards, consisting of 13 each of four suits: spades, hearts, diamonds, and clubs. Hearts and diamonds are red, while spades and clubs are black. Each suit has an ace, nine cards number 2 through 10, and three face cards. The face cards are a jack, a queen, and a king. Answers the following questions for a single card drawn at random from a well-shuffled deck of cards.

a. What is the probability of drawing a king of any suit?

b. What is the probability of drawing a face card that is also a spade?

c. What is the probability of drawing a card without a number on it?

d. What is the probability of drawing a red card? What is the probability of a drawing an ace? What is the probability of drawing a red ace? Are these events (“ace” and “red”) mutually exclusive? Are they independent?

e. List two events that are mutually exclusive for a single draw from a deck of cards.

f. What is the probability of drawing a red king? What is the probability of drawing a face card in hearts? Are these two events mutually exclusive? Are they independent?

Solution

a)

There are 4 kings out of 52 cards. Thus,

P(King) = 4/52 = 1/13 = 0.076923077

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b.

In spades, there are 3 face cards.

Thus,

P(face and spades) = 3/52 = 0.057692308

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c.

A, J, Q, K have no numbers on them. There are 16 of them.

Thus,

P(no number) = 16/52 = 4/13 = 0.307692308

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d.

There are 26 red cards. So,

P(red) = 26/52 = 1/2 = 0.5

There are 4 aces. Thus,

P(ace) = 4/52 = 1/13 = 0.076923077

They are no mutually exclusive as some aces are red.

They are independent, as the probability of getting an ace given it is red is just the same as when compared to all cards. The probability of a red given it is an ace is also just the same as when compared to all cards.
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