Questions: Drawing Cards If two cards are selected from a standard deck of 52 cards and are not replaced after each draw, find these probabilities. a. Both are 9 s . b. Both cards are the same suit. c. Both cards are spades.

Drawing Cards If two cards are selected from a standard deck of 52 cards and are not replaced after each draw, find these probabilities. a. Both are 9 s . b. Both cards are the same suit. c. Both cards are spades.
Transcript text: Drawing Cards If two cards are selected from a standard deck of 52 cards and are not replaced after each draw, find these probabilities. a. Both are 9 s . b. Both cards are the same suit. c. Both cards are spades.
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Solution

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Solution Steps

To solve these probability questions, we need to consider the total number of possible outcomes and the number of favorable outcomes for each scenario.

a. For both cards being 9s, we need to calculate the probability of drawing a 9 on the first draw and then another 9 on the second draw without replacement.

b. For both cards being the same suit, we need to calculate the probability of drawing a card of any suit on the first draw and then another card of the same suit on the second draw without replacement.

c. For both cards being spades, we need to calculate the probability of drawing a spade on the first draw and then another spade on the second draw without replacement.

Step 1: Calculate the Probability of Both Cards Being 9s

To find the probability that both cards drawn are 9s, we use the formula for conditional probability:

\[ P(\text{both 9s}) = \frac{\text{number of 9s}}{\text{total cards}} \times \frac{\text{remaining 9s}}{\text{remaining cards}} \]

Given:

  • Total cards = 52
  • Number of 9s = 4

\[ P(\text{both 9s}) = \frac{4}{52} \times \frac{3}{51} \approx 0.004524886877828055 \]

Step 2: Calculate the Probability of Both Cards Being the Same Suit

To find the probability that both cards drawn are of the same suit, we use the formula for conditional probability:

\[ P(\text{same suit}) = \frac{\text{cards per suit}}{\text{total cards}} \times \frac{\text{remaining cards of the same suit}}{\text{remaining cards}} \]

Given:

  • Total cards = 52
  • Cards per suit = 13

\[ P(\text{same suit}) = \frac{13}{52} \times \frac{12}{51} \approx 0.058823529411764705 \]

Step 3: Calculate the Probability of Both Cards Being Spades

To find the probability that both cards drawn are spades, we use the formula for conditional probability:

\[ P(\text{both spades}) = \frac{\text{number of spades}}{\text{total cards}} \times \frac{\text{remaining spades}}{\text{remaining cards}} \]

Given:

  • Total cards = 52
  • Number of spades = 13

\[ P(\text{both spades}) = \frac{13}{52} \times \frac{12}{51} \approx 0.058823529411764705 \]

Final Answer

\[ \boxed{P(\text{both 9s}) \approx 0.004524886877828055} \] \[ \boxed{P(\text{same suit}) \approx 0.05882} \] \[ \boxed{P(\text{both spades}) \approx 0.05882} \]

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