Questions: Here is a table showing all 52 cards in a standard deck. Suppose one card is drawn at random from a standard deck. Answer each part. Write your answers as fractions. (a) What is the probability that the card drawn is a diamond? (b) What is the probability that the card drawn is a face card? (c) What is the probability that the card drawn is a diamond or a face card?

Here is a table showing all 52 cards in a standard deck. Suppose one card is drawn at random from a standard deck. Answer each part. Write your answers as fractions. (a) What is the probability that the card drawn is a diamond? (b) What is the probability that the card drawn is a face card? (c) What is the probability that the card drawn is a diamond or a face card?
Transcript text: Here is a table showing all 52 cards in a standard deck. Suppose one card is drawn at random from a standard deck. Answer each part. Write your answers as fractions. (a) What is the probability that the card drawn is a diamond? $\square$ (b) What is the probability that the card drawn is a face card? $\square$ (c) What is the probability that the card drawn is a diamond or a face card? $\square$
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Solution

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Solution Steps

To solve these probability questions, we need to understand the composition of a standard deck of cards. A standard deck has 52 cards, consisting of 4 suits (hearts, diamonds, clubs, spades) with 13 cards each. Face cards are the Jack, Queen, and King in each suit, totaling 12 face cards.

(a) To find the probability of drawing a diamond, we calculate the ratio of diamond cards to the total number of cards.

(b) To find the probability of drawing a face card, we calculate the ratio of face cards to the total number of cards.

(c) To find the probability of drawing a diamond or a face card, we use the principle of inclusion-exclusion. We add the probabilities of drawing a diamond and a face card, then subtract the probability of drawing a card that is both a diamond and a face card.

Step 1: Probability of Drawing a Diamond

The probability of drawing a diamond from a standard deck of cards is given by the ratio of the number of diamond cards to the total number of cards. There are 13 diamonds in a deck of 52 cards. Thus, the probability is:

\[ P(\text{Diamond}) = \frac{13}{52} = \frac{1}{4} = 0.25 \]

Step 2: Probability of Drawing a Face Card

The probability of drawing a face card is calculated by the ratio of the number of face cards to the total number of cards. There are 12 face cards in a deck of 52 cards. Therefore, the probability is:

\[ P(\text{Face Card}) = \frac{12}{52} = \frac{3}{13} \approx 0.2308 \]

Step 3: Probability of Drawing a Diamond or a Face Card

To find the probability of drawing a card that is either a diamond or a face card, we use the principle of inclusion-exclusion. We add the probabilities of drawing a diamond and a face card, then subtract the probability of drawing a card that is both a diamond and a face card (which are the 3 face cards that are diamonds):

\[ P(\text{Diamond or Face Card}) = P(\text{Diamond}) + P(\text{Face Card}) - P(\text{Diamond and Face Card}) \]

Substituting the values:

\[ P(\text{Diamond or Face Card}) = \frac{1}{4} + \frac{3}{13} - \frac{3}{52} \]

Calculating this gives:

\[ P(\text{Diamond or Face Card}) = 0.25 + 0.2308 - 0.0577 \approx 0.4231 \]

Final Answer

The probabilities are as follows:

  • (a) \( P(\text{Diamond}) = \frac{1}{4} \)
  • (b) \( P(\text{Face Card}) = \frac{3}{13} \)
  • (c) \( P(\text{Diamond or Face Card}) \approx 0.4231 \)

Thus, the final answers are: \[ \boxed{P(\text{Diamond}) = \frac{1}{4}, \; P(\text{Face Card}) = \frac{3}{13}, \; P(\text{Diamond or Face Card}) \approx 0.4231} \]

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