Question

determine whether the following individual events are overlapping or non-overlapping. then find the probability of the combined event.

drawing either a 5 or a spade from a regular deck of cards

select the correct choice below and fill in the answer box to complete your choice.
(type an integer or a simplified fraction.)

a. the individual events are overlapping. the probability of the combined event is
b. the individual events are non-overlapping. the probability of the combined event is

Explanation:

Determine if the events overlap

The two individual events are:

1. Drawing a 5.
2. Drawing a spade.

Since the 5 of spades is both a 5 and a spade, these two events can occur at the same time. Therefore, the events are overlapping.

Calculate the individual and overlapping probabilities

A standard deck has 52 cards.

• Number of 5s: \(4\)
• Number of spades: \(13\)
• Number of cards that are both a 5 and a spade (5 of spades): \(1\)

$$ P(\text{5}) = \frac{4}{52} $$

$$ P(\text{spade}) = \frac{13}{52} $$

$$ P(\text{5 and spade}) = \frac{1}{52} $$

Calculate the combined probability

Using the addition rule for overlapping events:

$$ P(\text{5 or spade}) = P(\text{5}) + P(\text{spade}) - P(\text{5 and spade}) $$

$$ P(\text{5 or spade}) = \frac{4}{52} + \frac{13}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13} $$

Answer:

• A. The individual events are overlapping. The probability of the combined event is \(\frac{4}{13}\). (Correct answer)
• B. The individual events are non-overlapping. The probability of the combined event is \(\frac{4}{13}\).