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 = 5/4 Odds against = 4/5 (B) Odds for = Odds against =

Solution

Given the following probabilities for an event \( E \), find the odds for and against \( E \). Let’s solve for option (A), where the probability of event \( E \) is \( \frac{5}{9} \). Odds for \( E \): The odds for an event can be found using the formula: \[ \text{Odds for } E = \frac{P(E)}{1-P(E)} \] Substitute the given probability: \[ \text{Odds for } E = \frac{\frac{5}{9}}{1 - \frac{5}{9}} \] \[ = \frac{\frac{5}{9}}{\frac{4}{9}} \] \[ = \frac{5}{4} \] Odds against \( E \): The odds against an event can be found using the formula: \[ \text{Odds against } E = \frac{1-P(E)}{P(E)} \] Substitute the given probability: \[ \text{Odds against } E = \frac{1 - \frac{5}{9}}{\frac{5}{9}} \] \[ = \frac{\frac{4}{9}}{\frac{5}{9}} \] \[ = \frac{4}{5} \] Repeat the same process for option (B), where the probability of event \( E \) is \( \frac{7}{8} \). Odds for \( E \): \[ \text{Odds for } E = \frac{\frac{7}{8}}{1 - \frac{7}{8}} \] \[ = \frac{\frac{7}{8}}{\frac{1}{8}} \] \[ = 7 \] Odds against \( E \): \[ \text{Odds against } E = \frac{1 - \frac{7}{8}}{\frac{7}{8}} \] \[ = \frac{\frac{1}{8}}{\frac{7}{8}} \] \[ = \frac{1}{7} \]