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 to . (simplify your answers.)

Explanation:

Identify outcomes of rolling a fair die

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

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

The total number of equally likely outcomes is:

$$ n(S) = 6 $$

Determine favorable and unfavorable outcomes

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

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

The number of favorable outcomes is:

$$ a = n(E) = 2 $$

The number of unfavorable outcomes (outcomes not in \(E\)) is:

$$ b = n(S) - n(E) = 6 - 2 = 4 $$

Calculate the simplified odds for the event

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

$$ \text{Odds for} = a : b = 2 : 4 $$

Simplifying the ratio by dividing both terms by their greatest common divisor, \(2\):

$$ \text{Odds for} = 1 : 2 $$

Thus, the odds for the event are \(1\) to \(2\).

Answer:

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 <blank>1</blank> to <blank>2</blank>. (Simplify your answers.)