Question: Given the following probabilities for an event E, find the odds for and against E. (A) 5/9 (B) 7/8…

Given the following probabilities for an event E, find the odds for and against E. (A) 5/9 (B) 7/8 (C) 0.17 (D) 0.49 For all answers, type the odds as a fraction. Simplify your answers. (A) Odds for = Odds against =

Solution

To solve the problem, we will need to calculate the odds for and against an event \( E \) using one of the given probabilities. Let’s choose option (A) \( \frac{5}{9} \) for this calculation. First, recall that the odds for an event happening are calculated as: \[ \text{Odds for} = \frac{P(E)}{1 - P(E)} \] Calculate \( 1 - P(E) \) for \( P(E) = \frac{5}{9} \): \[ 1 - P(E) = 1 - \frac{5}{9} = \frac{9}{9} - \frac{5}{9} = \frac{4}{9} \] Now, compute the odds for: \[ \text{Odds for} = \frac{\frac{5}{9}}{\frac{4}{9}} = \frac{5}{4} \] Next, compute the odds against which are the reciprocal of the odds for: \[ \text{Odds against} = \frac{1 - P(E)}{P(E)} = \frac{\frac{4}{9}}{\frac{5}{9}} = \frac{4}{5} \] Therefore, the odds for the event \( E \) are \( \frac{5}{4} \) and the odds against the event \( E \) are \( \frac{4}{5} \).