Question

in each case, find the probability of an event e having the given odds.
(a) the odds in favor of e are 5 to 7.
(b) the odds against e are 8 to 10.

(a) the probability of an event e occurring is \\(\frac{5}{12}\\)
(type an integer or a simplified fraction.)
(b) the probability of an event e occurring is \\(square\\).
(type an integer or a simplified fraction.)

Explanation:

Step1: Recall the formula for probability from odds against

If the odds against an event \( E \) are \( a \) to \( b \), then the probability of \( E \) occurring is \( \frac{b}{a + b} \). Here, the odds against \( E \) are 8 to 10, so \( a = 8 \) and \( b = 10 \).

Step2: Calculate the probability

Substitute \( a = 8 \) and \( b = 10 \) into the formula. The total number of outcomes is \( a + b = 8 + 10 = 18 \). The number of favorable outcomes for \( E \) is \( b = 10 \). So the probability \( P(E)=\frac{10}{18} \), which simplifies to \( \frac{5}{9} \) by dividing both the numerator and denominator by 2.

Answer:

\(\frac{5}{9}\)