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 A card is drawn from a standard deck of 52 cards. 

 a. What is the probability the 7 of spades is drawn?

b. What is the probability that a 7 is drawn?

 c. What is the probability that a face card is drawn?

d. What is the probability that a heart is drawn?

e. What are the odds that a heart is drawn?

 f. What are the odds that a king or queen is drawn?

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 a. What is the probability the 7 of spades is drawn?

Let \(\mathrm{N}\) be the number of cards in a deck of cards.
Let \(S\) be the number of "seven of spades" in the deck.
The probability of drawing the "seven of spades" is \(S / N\)
If there are 52 cards then \(N=52\).
If there is 1 "seven of spades" in the deck then \(S=1\).
Then the probability of drawing the "seven of spades" from a deck of cards is \(1 / 52\) or \(0.01923\) or \(1.923 \%\)

 

b. What is the probability that a 7 is drawn?

There are four 7s in a standard deck, and there are a total of 52 cards. So:

\[P(7) = \dfrac{4}{52} =\dfrac{1}{13}\]

 

c. What is the probability that a face card is drawn?

\[P(Face_Card) = \dfrac{12}{52}=\dfrac{3}{13}\]

 

d. What is the probability that a heart is drawn?

A standard deck contains an equal number of hearts, diamonds, clubs, and spades. So the probability of drawing a heart is:

\[P(Heart)=\dfrac{13}{52}=\dfrac{1}{4}\]

 

e. What are the odds that a heart is drawn?

The odds in favor are expressed as [the number of favorable outcomes]:[the number of unfavorable outcomes] and then divide out any common factors.

In a 52 card deck there are 4 suits, so 52 divided by 4 is 13 -- hence there are 13 hearts. 52 minus 13 is 39, so there are 39 cards that are not hearts. So if drawing a heart is considered a favorable outcome, then there are 13 possible favorable outcomes, and there are 39 possible unfavorable outcomes. Hence, the odds in favor of drawing a heart are 13 to 39. But notice that both 13 and 39 are evenly divisible by 13, so reduced to lowest terms we have 1 to 3.

 

f. What are the odds that a king or queen is drawn?

8 kings and queens to 44 non-kings and non queens \(=8: 44\) which reduces to \(2: 11\)

 

 

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