Question

texes mathematics 4-8 (115)

1. three cards are randomly drawn from a deck of playing cards in succession without replacement. what is the probability that the first is a king, the second is an ace, and the third is a queen?

a. \\(\frac{8}{725}\\)
b. \\(\frac{8}{16575}\\)
c. \\(\frac{7}{16575}\\)
d. \\(\frac{11}{100}\\)

1. a card is drawn from an ordinary deck of cards. what is the probability of drawing a number card (not including the ace) or a king?

a. \\(\frac{4}{13}\\)
b. \\(\frac{1}{52}\\)
c. \\(\frac{13}{52}\\)
d. \\(\frac{10}{13}\\)

Explanation:

🆕 New Concept Discovered: Dependent Probability
Calculating likelihood when previous outcomes affect the next.

Step 1: Identify the card counts for each event

A standard deck has \( 52 \) cards in total.

• Number of Kings: \( 4 \)
• Number of Aces: \( 4 \)
• Number of Queens: \( 4 \)

Step 2: Calculate the probability of each sequential draw

Since the cards are drawn without replacement, the total number of cards decreases by \( 1 \) after each draw:

1. First card is a King:

$$ P(\text{King}) = \frac{4}{52} $$

1. Second card is an Ace (given the first was a King):

There are still \( 4 \) Aces left, but only \( 51 \) cards remaining in the deck.

$$ P(\text{Ace} \mid \text{King}) = \frac{4}{51} $$

1. Third card is a Queen (given the first two were a King and an Ace):

There are still \( 4 \) Queens left, but only \( 50 \) cards remaining in the deck.

$$ P(\text{Queen} \mid \text{King and Ace}) = \frac{4}{50} $$

Step 3: Multiply the probabilities

Multiply the probabilities of these dependent events to find the joint probability:

$$ P(\text{King, then Ace, then Queen}) = \frac{4}{52} \times \frac{4}{51} \times \frac{4}{50} $$

Simplify the individual fractions before multiplying:

• \(\frac{4}{52} = \frac{1}{13}\)
• \(\frac{4}{50} = \frac{2}{25}\)

Now, multiply the simplified fractions:

$$ P = \frac{1}{13} \times \frac{4}{51} \times \frac{2}{25} $$

$$ P = \frac{1 \times 4 \times 2}{13 \times 51 \times 25} $$

$$ P = \frac{8}{16575} $$

Answer:

B. \(\frac{8}{16575}\)