Questions: Sara draws the 2 of hearts from a standard deck of 52 cards. Without replacing the first card, she then proceeds to draw a second card. a. Determine the probability that the second card is another 2 . P(2 2 of hearts) = b. Determine the probability that the second card is another heart. P(heart 2 of hearts) = c. Determine the probability that the second card is a club. P(club 2 of hearts) = d. Determine the probability that the second card is a 9. P(9 2 of hearts) = Write your answers as reduced fractions.

Sara draws the 2 of hearts from a standard deck of 52 cards. Without replacing the first card, she then proceeds to draw a second card. a. Determine the probability that the second card is another 2 . P(2 2 of hearts) = b. Determine the probability that the second card is another heart. P(heart 2 of hearts) = c. Determine the probability that the second card is a club. P(club 2 of hearts) = d. Determine the probability that the second card is a 9. P(9 2 of hearts) = Write your answers as reduced fractions.
Transcript text: Sara draws the 2 of hearts from a standard deck of 52 cards. Without replacing the first card, she then proceeds to draw a second card. a. Determine the probability that the second card is another 2 . \[ P(2 \mid 2 \text { of hearts })= \] $\square$ b. Determine the probability that the second card is another heart. \[ P(\text { heart } \mid 2 \text { of hearts })= \] $\square$ c. Determine the probability that the second card is a club. \[ P(\text { club } \mid 2 \text { of hearts })= \] $\square$ d. Determine the probability that the second card is a 9. \[ P(9 \mid 2 \text { of hearts })= \] $\square$ Write your answers as reduced fractions. Calculator Submit Question
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Solution

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Solution Steps

To solve these probability questions, we need to consider the total number of cards left in the deck after the first card is drawn and the number of favorable outcomes for each event.

a. After drawing the 2 of hearts, there are 51 cards left in the deck. There are 3 remaining 2s in the deck. b. After drawing the 2 of hearts, there are 51 cards left in the deck. There are 12 remaining hearts in the deck. c. After drawing the 2 of hearts, there are 51 cards left in the deck. There are 13 clubs in the deck.

Step 1: Probability that the second card is another 2

After drawing the 2 of hearts, there are 51 cards left in the deck. The number of favorable outcomes for drawing another 2 is 3 (the 2 of diamonds, 2 of clubs, and 2 of spades). Therefore, the probability is given by:

\[ P(2 \mid 2 \text{ of hearts}) = \frac{3}{51} = \frac{1}{17} \]

Step 2: Probability that the second card is another heart

With the 2 of hearts already drawn, there are still 12 hearts remaining in the deck of 51 cards. Thus, the probability that the second card is a heart is:

\[ P(\text{heart} \mid 2 \text{ of hearts}) = \frac{12}{51} = \frac{4}{17} \]

Step 3: Probability that the second card is a club

After drawing the 2 of hearts, there are still 13 clubs in the remaining 51 cards. Therefore, the probability that the second card is a club is:

\[ P(\text{club} \mid 2 \text{ of hearts}) = \frac{13}{51} \]

Final Answer

\[ \boxed{ \begin{align_} P(2 \mid 2 \text{ of hearts}) & = \frac{1}{17} \\ P(\text{heart} \mid 2 \text{ of hearts}) & = \frac{4}{17} \\ P(\text{club} \mid 2 \text{ of hearts}) & = \frac{13}{51} \end{align_} } \]

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