Question

event a or b: rolling an odd number or rolling a number less than 3

event a and b: rolling an odd number and rolling a number less than 3

(b) compute the following.

\\(p(a) + p(b) - p(a \text{ and } b) = \\)

(c) select the answer that makes the equation true.

\\(p(a) + p(b) - p(a \text{ and } b) = p(a \text{ or } b)\\)

Explanation:

Define the sample space and events

The experiment is rolling a standard six-sided die:

$$ S = \{1, 2, 3, 4, 5, 6\} $$

Event \(A\) is rolling an odd number:

$$ A = \{1, 3, 5\} \implies P(A) = \frac{3}{6} = \frac{1}{2} $$

Event \(B\) is rolling a number less than 3:

$$ B = \{1, 2\} \implies P(B) = \frac{2}{6} = \frac{1}{3} $$

Find the intersection of events

Event \(A \text{ and } B\) is rolling an odd number and a number less than 3:

$$ A \cap B = \{1\} \implies P(A \text{ and } B) = \frac{1}{6} $$

Calculate the required expression

Substitute the probabilities into the formula:

$$ P(A) + P(B) - P(A \text{ and } B) = \frac{3}{6} + \frac{2}{6} - \frac{1}{6} = \frac{4}{6} = \frac{2}{3} $$

By the addition rule of probability, this expression is equal to \(P(A \text{ or } B)\).

Answer:

Question 1

\(P(A) + P(B) - P(A \text{ and } B) = \frac{2}{3}\)

Question 2

\(P(A) + P(B) - P(A \text{ and } B) = P(A \text{ or } B)\)