An experiment is picking a card from a fair deck.
a) What is the probability of picking a King?

Round answer to 3 decimal places.

b) What is the probability of picking a King given that the card is a face card?

Round answer to 3 decimal places.

c) What is the probability of picking a face card given that the card is a King?

Round answer to 3 decimal places.

d) Are the events King and face card independent events? Why or why not?

Round answer to 3 decimal places.

Ask by Simpson Warner. in the United States

Mar 10,2025

Question

An experiment is picking a card from a fair deck.
a) What is the probability of picking a King?

Round answer to 3 decimal places.

b) What is the probability of picking a King given that the card is a face card?

Round answer to 3 decimal places.

c) What is the probability of picking a face card given that the card is a King?

Round answer to 3 decimal places.

d) Are the events King and face card independent events? Why or why not?

Round answer to 3 decimal places.

Ask by Simpson Warner. in the United States

Mar 10,2025

UpStudy AI · Accepted Answer

a) There are 4 Kings in a deck of 52 cards. Thus, the probability of picking a King is  
$$
P(\text{King})=\frac{4}{52}=\frac{1}{13}\approx0.077.
$$

b) The face cards in a deck are Jacks, Queens, and Kings. There are 12 in total (4 of each). Given that the card is a face card, the probability that it is a King is  
$$
P(\text{King} \mid \text{face card})=\frac{4}{12}=\frac{1}{3}\approx0.333.
$$

c) Since every King is a face card, if the card is a King then it is certainly a face card. Therefore,  
$$
P(\text{face card} \mid \text{King})=1.000.
$$

d) Two events $$A$$ and $$B$$ are independent if and only if  
$$
P(A\text{ and }B)=P(A)\times P(B).
$$  
Here, let $$A$$ be the event of drawing a King and $$B$$ be the event of drawing a face card. We have:
$$
P(\text{King})=\frac{4}{52}=\frac{1}{13},\quad P(\text{face card})=\frac{12}{52}=\frac{3}{13},
$$
and 
$$
P(\text{King and face card})=\frac{4}{52}=\frac{1}{13}.
$$
The product is 
$$
P(\text{King})\times P(\text{face card})=\frac{1}{13}\times\frac{3}{13}=\frac{3}{169}\approx0.0178,
$$
which is not equal to 
$$
P(\text{King and face card})\approx0.0769.
$$  
Since $$\frac{3}{169}\neq\frac{1}{13}$$, the events are not independent.
a) The probability of picking a King is 0.077. b) The probability of picking a King given that the card is a face card is 0.333. c) The probability of picking a face card given that the card is a King is 1.000. d) The events King and face card are not independent because the probability of both events occurring together does not equal the product of their individual probabilities.

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