Question

1. 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. let’s 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|h) ).
(round 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}) ).
(round to 3 decimal places. leave your answer in decimal form.)

6c.1 find ( p(f \text{ or } h) ).
(round to 3 decimal places. leave your answer in decimal form.)

Explanation:

Step1: Find \( P(F \cap f) \)

The formula for probability \( P(A)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\).
For \( P(F \cap f) \), the number of favorable outcomes (face - card and heart) is \( 3 \), and the total number of outcomes is \( 52 \).
So, \( P(F \cap f)=\frac{3}{52}\approx0.06\).

Step2: Describe the probability in words

The probability that a card is a face - card and a heart from this deck.

Step3: Find \( P(F|ace \ card) \)

The formula for conditional probability \( P(A|B)=\frac{P(A\cap B)}{P(B)}\).
Here, \( A = F \) (face - card) and \( B=\text{ace card}\). \( P(F\cap\text{ace card}) = 0\) (since an ace is not a face - card). So \( P(F|\text{ace card})=\frac{0}{4}=0\).

Step4: Find \( P(face \ card \ and \ next \ a \ heart) \)

We use the formula for conditional probability \( P(A\cap B)=P(A)\times P(B|A)\).
The probability of drawing a face - card \( P(A)=\frac{12}{52}\). After drawing a face - card, there are \( 51\) cards left. The number of non - face heart cards is \( 10\). So \( P(B|A)=\frac{10}{51}\).
Then \( P(\text{face card and next a heart})=\frac{12}{52}\times\frac{10}{51}=\frac{120}{2652}\approx0.045\).

Answer:

(a) \( 0.06\)
(b) \( 0.045\)