Question

1. drawing a card if one card is drawn from a deck, find the probability of getting these results: a. a 3 or a diamond b. a red six c. a black card higher than a 10 (dont count aces) d. a spade e. a red card

Explanation:

Step1: Define total cards

A standard deck has $n(S) = 52$ cards.
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Part a: 3 or a diamond

Step1: Count favorable cards

Number of 3s: 4; Number of diamonds: 13; Number of 3 of diamonds (overlap): 1. Use addition rule: $n(3 \cup \text{diamond}) = 4 + 13 - 1 = 16$

Step2: Calculate probability

$P(3 \cup \text{diamond}) = \frac{n(3 \cup \text{diamond})}{n(S)} = \frac{16}{52} = \frac{4}{13}$
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Part b: A red six

Step1: Count favorable cards

There are 2 red sixes (6 of hearts, 6 of diamonds): $n(\text{red six}) = 2$

Step2: Calculate probability

$P(\text{red six}) = \frac{2}{52} = \frac{1}{26}$
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Part c: Black card >10 (no aces)

Step1: Count favorable cards

Black suits (spades, clubs) have J, Q, K as cards >10: $3 \times 2 = 6$ cards

Step2: Calculate probability

$P(\text{black >10}) = \frac{6}{52} = \frac{3}{26}$
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Part d: A spade

Step1: Count favorable cards

There are 13 spades in a deck: $n(\text{spade}) = 13$

Step2: Calculate probability

$P(\text{spade}) = \frac{13}{52} = \frac{1}{4}$
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Part e: A red card

Step1: Count favorable cards

Half the deck is red: $n(\text{red card}) = 26$

Step2: Calculate probability

$P(\text{red card}) = \frac{26}{52} = \frac{1}{2}$

Answer:

a. $\frac{4}{13}$
b. $\frac{1}{26}$
c. $\frac{3}{26}$
d. $\frac{1}{4}$
e. $\frac{1}{2}$