Question

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

drawing either a 3 or a red card 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 are overlapping

Using the Overlapping Events knowledge point

• Let Event \(A\) be drawing a 3.
• Let Event \(B\) be drawing a red card.
• A standard deck of 52 cards contains red 3s (the 3 of hearts and the 3 of diamonds).
• Since a card can be both a 3 and a red card, the events share common outcomes.
• Therefore, the individual events are overlapping.

Calculate the individual probabilities

Using the Probability knowledge point

• Total cards in a standard deck: \(N(S) = 52\).
• Number of 3s: \(N(A) = 4\).
• Number of red cards: \(N(B) = 26\).
• Number of cards that are both a 3 and red: \(N(A \cap B) = 2\).

The individual probabilities are:

$$ P(A) = \frac{4}{52}, \quad P(B) = \frac{26}{52}, \quad P(A \cap B) = \frac{2}{52} $$

Calculate the combined probability

Using the Overlapping Events and Probability knowledge points

• Apply the addition rule for overlapping events:

$$ P(A \cup B) = P(A) + P(B) - P(A \cap B) $$

• Substitute the values:

$$ P(A \cup B) = \frac{4}{52} + \frac{26}{52} - \frac{2}{52} = \frac{28}{52} $$

• Simplify the fraction:

$$ P(A \cup B) = \frac{28 \div 4}{52 \div 4} = \frac{7}{13} $$

Answer:

• A. The individual events are overlapping. The probability of the combined event is <blank>\(\frac{7}{13}\)</blank> (Correct answer)
• B. The individual events are non-overlapping. The probability of the combined event is <blank>\(\frac{15}{26}\)</blank>