Math / Data & Statistics

QuestionQuestion 4 1 pts
Solve the problem involving probabilities with independent events.
You are dealt one card from a 52 card deck. Then the card is replaced in the deck, the deck is shuffled, and you draw again. Find the probability of getting a picture card the first time and a club the second time. $\frac{3}{52}$ $\frac{1}{13}$ $\frac{1}{4}$ $\frac{3}{13}$

Studdy Solution

STEP 1

1. A standard deck of 52 cards is used.
2. Picture cards refer to Jacks, Queens, and Kings.
3. There are 12 picture cards in a deck (3 per suit).
4. There are 13 clubs in a deck.
5. The events are independent because the card is replaced and the deck is shuffled before the second draw.

STEP 2

1. Calculate the probability of drawing a picture card on the first draw.
2. Calculate the probability of drawing a club on the second draw.
3. Multiply the probabilities of the two independent events.

STEP 3

Calculate the probability of drawing a picture card on the first draw.
There are 12 picture cards in a deck of 52 cards.
Probability of drawing a picture card:
$P(\text{Picture card}) = \frac{12}{52} = \frac{3}{13}$

STEP 4

Calculate the probability of drawing a club on the second draw.
There are 13 clubs in a deck of 52 cards.
Probability of drawing a club:
$P(\text{Club}) = \frac{13}{52} = \frac{1}{4}$

STEP 5

Multiply the probabilities of the two independent events.
$P(\text{Picture card first and Club second}) = P(\text{Picture card}) \times P(\text{Club})$
$P(\text{Picture card first and Club second}) = \frac{3}{13} \times \frac{1}{4} = \frac{3}{52}$
The probability of getting a picture card the first time and a club the second time is:
$\boxed{\frac{3}{52}}$

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