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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

Solución de inteligencia artificial de Upstudy

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Responder

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.

Solución

a) There are 4 Kings in a deck of 52 cards. Thus, the probability of picking a King is
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
c) Since every King is a face card, if the card is a King then it is certainly a face card. Therefore,
d) Two events and are independent if and only if
Here, let be the event of drawing a King and be the event of drawing a face card. We have:
and
The product is
which is not equal to
Since , the events are not independent.

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Mind Expander

The experiment of picking a card from a fair deck is one of those classic probability problems that quite literally stacks the odds in your favor of learning something new! A standard deck contains 52 cards, with 4 Kings and a total of 12 face cards (Kings, Queens, and Jacks). So, you can easily figure out the probabilities by remembering that math is just like a game—sometimes you score big, and sometimes you just learn a lesson!
Believe it or not, the concept of independence in probability can be like playing detective! Events are independent when the occurrence of one doesn’t affect the probability of another. In this case, since knowing a card is a King means it is also a face card, the two events are not independent. If you find a King, it guarantees a face card, making those two events way too intertwined, just like a classic whodunit plot twist!

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