Question

question 26

suppose events e and f are mutually exclusive, with p(f)=0.66 and p(e∪f)=0.84. calculate the following and enter your answer one digit per box (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 mutually - exclusive events

For mutually - exclusive events \(E\) and \(F\), \(P(E\cup F)=P(E)+P(F)\). Given \(P(F) = 0.66\) and \(P(E\cup F)=0.84\), we can find \(P(E)\) as \(P(E)=P(E\cup F)-P(F)\).

$$P(E)=0.84 - 0.66=0.18$$

Step2: Calculate the probability of the complement of \(E\)

The probability of the complement of \(E\), \(P(\overline{E})=1 - P(E)\).

$$P(\overline{E})=1 - 0.18 = 0.82$$

Step3: Calculate the odds for \(E\)

The odds for an event \(E\) is given by \(\frac{P(E)}{P(\overline{E})}\).

$$\frac{P(E)}{P(\overline{E})}=\frac{0.18}{0.82}=\frac{18}{82}=\frac{9}{41}$$

So the odds for \(E\) are \(9\) to \(41\).

Step4: Calculate the odds against \(E\)

The odds against an event \(E\) is given by \(\frac{P(\overline{E})}{P(E)}\).

$$\frac{P(\overline{E})}{P(E)}=\frac{0.82}{0.18}=\frac{82}{18}=\frac{41}{9}$$

So the odds against \(E\) are \(41\) to \(9\).

Answer:

a) \(9\) to \(41\)
b) \(41\) to \(9\)