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 19

(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 \( P(E)=\frac{b}{a + b} \). Here, the odds against \( E \) are 8 to 19, so \( a = 8 \) and \( b=19 \).

Step2: Calculate the probability

Substitute \( a = 8 \) and \( b = 19 \) into the formula: \( P(E)=\frac{19}{8 + 19}=\frac{19}{27} \).

Answer:

\(\frac{19}{27}\)