Question

find the odds for and the odds against the event rolling a fair die and getting a 4 or a 5.

a. the odds for the event are 1 to 2. (simplify your answers.)
b. the odds against the event are to . (simplify your answers.)

Explanation:

Identify outcomes of rolling a fair die

The sample space of rolling a fair six-sided die is:

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

The event \(E\) of getting a 4 or a 5 is:

$$ E = \{4, 5\} $$

The number of favorable outcomes is:

$$ n(E) = 2 $$

The number of unfavorable outcomes is:

$$ n(E') = 6 - 2 = 4 $$

Calculate the odds for the event

The odds for the event are given by the ratio of favorable outcomes to unfavorable outcomes:

$$ \text{Odds for} = n(E) : n(E') = 2 : 4 = 1 : 2 $$

Calculate the odds against the event

The odds against the event are given by the ratio of unfavorable outcomes to favorable outcomes:

$$ \text{Odds against} = n(E') : n(E) = 4 : 2 = 2 : 1 $$

Answer:

Question 1

a. The odds for the event are <blank>1</blank> to <blank>2</blank>. (Simplify your answers.)

Question 2

b. The odds against the event are <blank>2</blank> to <blank>1</blank>. (Simplify your answers.)