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QuestionFind the probabilities using a 52-card deck. Reduce your answers to fractions.
a. Probability of drawing a diamond: P(D)=1352P(D) = \frac{13}{52}.
b. Probability of drawing a king then a 2: P(K,2)=452⋅451P(K, 2) = \frac{4}{52} \cdot \frac{4}{51}.
c. Probability of drawing a heart then a queen: P(H,Q)=1352⋅451P(H, Q) = \frac{13}{52} \cdot \frac{4}{51}.

Studdy Solution

STEP 1

Assumptions1. A standard deck of52 cards is used. . The deck contains4 suits hearts, diamonds, clubs, and spades.
3. Each suit has13 cards Ace,,3,4,5,6,7,8,9,10, Jack, Queen, and King.
4. The cards are well shuffled, so each card has an equal chance of being drawn.
5. When two cards are drawn, the first card is not replaced before drawing the second card.

STEP 2

First, we need to find the probability that a randomly selected card is a diamond. The probability of an event is calculated as the number of favorable outcomes divided by the total number of outcomes.
(Diamond)=Number of diamondsTotal number of cards(Diamond) = \frac{Number\, of\, diamonds}{Total\, number\, of\, cards}

STEP 3

Now, plug in the known values. There are13 diamonds in a deck of52 cards.
(Diamond)=1352(Diamond) = \frac{13}{52}

STEP 4

Calculate the probability.
(Diamond)=1352=14(Diamond) = \frac{13}{52} = \frac{1}{4}

STEP 5

Next, we need to find the probability of drawing a king first, followed by drawing a2. Since these are two independent events, we can multiply their individual probabilities to get the overall probability.
(King then 2)=(King)×(2∣King)(King\, then\,2) =(King) \times(2|King)

STEP 6

Now, plug in the known values. There are4 kings and4 twos in a deck of52 cards. After drawing a king, there are51 cards left.
(King then 2)=452×451(King\, then\,2) = \frac{4}{52} \times \frac{4}{51}

STEP 7

Calculate the probability.
(King then 2)=452×451=1169(King\, then\,2) = \frac{4}{52} \times \frac{4}{51} = \frac{1}{169}

STEP 8

Finally, we need to find the probability of drawing a heart first, followed by drawing a queen. Again, we multiply the individual probabilities.
(Heart then Queen)=(Heart)×(Queen∣Heart)(Heart\, then\, Queen) =(Heart) \times(Queen|Heart)

STEP 9

Now, plug in the known values. There are13 hearts and4 queens in a deck of52 cards. After drawing a heart, there are51 cards left.
(Heart then Queen)=1352×451(Heart\, then\, Queen) = \frac{13}{52} \times \frac{4}{51}

STEP 10

Calculate the probability.
(Heart then Queen)=1352×451=52(Heart\, then\, Queen) = \frac{13}{52} \times \frac{4}{51} = \frac{}{52}The probabilities are as followsa. The probability that a randomly selected card is a diamond is/4. b. The probability of drawing a king first, followed by drawing a2 is/169. c. The probability of drawing a heart first, followed by drawing a queen is/52.

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