Question

question 24

suppose events e and f are independent with ( p ( f ) = 0.72 ) and ( p ( e cap f ) = 0.288 ). calculate the following and enter your answers one digit per box (please simplify your numbers to the extent possible).

a) the odds for e are: to

b) the odds against e are: to

Explanation:

Step1: Use the formula for independent events

For independent events \(E\) and \(F\), \(P(E\cap F)=P(E)\times P(F)\). Given \(P(F) = 0.72\) and \(P(E\cap F)=0.288\), we can find \(P(E)\) by \(P(E)=\frac{P(E\cap F)}{P(F)}\).

$$P(E)=\frac{0.288}{0.72}=0.4$$

Step2: Calculate the odds for \(E\)

The odds for an event \(E\) is given by \(\frac{P(E)}{1 - P(E)}\). Substituting \(P(E)=0.4\), we get \(\frac{0.4}{1 - 0.4}=\frac{0.4}{0.6}=\frac{2}{3}\). So the odds for \(E\) are \(2\) to \(3\).

Step3: Calculate the odds against \(E\)

The odds against an event \(E\) is given by \(\frac{1 - P(E)}{P(E)}\). Substituting \(P(E)=0.4\), we get \(\frac{1 - 0.4}{0.4}=\frac{0.6}{0.4}=\frac{3}{2}\). So the odds against \(E\) are \(3\) to \(2\).

Answer:

a) \(2\) to \(3\)
b) \(3\) to \(2\)