Question

a standard deck of playing cards (with jokers removed) consists of 52 cards in four suits - clubs, diamonds, hearts, and spades. each suit has 13 cards, with denominations ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, and king. the jack, queen, and king are referred to as \face cards.\ imagine that we shuffle the deck thoroughly and deal one card. lets define events f: getting a face card and h: getting a heart. the two - way table summarizes the sample space for this chance process.

(a) find ( p(f \text{ if } f) ).

(record to 2 decimal places. leave your answer in decimal form)

describe this probability in words.

there is a probability that a from this deck a heart.

(b) find ( p(\text{face card and not a heart}) ).

(record to 3 decimal places. leave your answer in decimal form)

(c) find ( p(f \text{ or } h) ).

(record to 3 decimal places. leave your answer in decimal form)

Explanation:

Step1: Use the formula for conditional probability

The formula for conditional probability is \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). Here, \(A\) is the event of getting a face - card and \(B\) is the event of getting a heart. \(P(F\cap H)=\frac{3}{52}\) (from the table, the number of face - cards that are hearts is 3 out of 52 cards) and \(P(H)=\frac{13}{52}\).

Step2: Calculate \(P(F|H)\)

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Step3: Use the formula for \(P(\text{face card and not a heart})\)

The number of face - cards that are not hearts is 9 (from the table). So \(P(\text{face card and not a heart})=\frac{9}{52}\approx0.173\)

Step4: Use the formula for \(P(H|\text{face card})\)

By the formula \(P(H|F)=\frac{P(F\cap H)}{P(F)}\). \(P(F)=\frac{12}{52}\) (total number of face - cards is 12) and \(P(F\cap H)=\frac{3}{52}\)

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Answer:

(a) \(0.231\) (b) \(0.173\) (c) \(0.25\)