Question

question 6
1 pts
suppose events e and f are mutually exclusive, with p(f)=0.54 and p(e∪f)=0.64. calculate the following and enter your answer 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: Find \(P(E)\)

Since \(E\) and \(F\) are mutually - exclusive, \(P(E\cup F)=P(E)+P(F)\).
We know \(P(E\cup F) = 0.64\) and \(P(F)=0.54\).
So, \(P(E)=P(E\cup F)-P(F)\)
\(P(E)=0.64 - 0.54=0.1\)

Step2: Calculate the odds for \(E\)

The formula for odds for an event \(E\) is \(\text{Odds for }E=\frac{P(E)}{1 - P(E)}\)
Substitute \(P(E) = 0.1\) into the formula:
\(\text{Odds for }E=\frac{0.1}{1 - 0.1}=\frac{0.1}{0.9}=\frac{1}{9}\)

Step3: Calculate the odds against \(E\)

The formula for odds against an event \(E\) is \(\text{Odds against }E=\frac{1 - P(E)}{P(E)}\)
Substitute \(P(E)=0.1\) into the formula:
\(\text{Odds against }E=\frac{1 - 0.1}{0.1}=\frac{0.9}{0.1}=\frac{9}{1}\)

Answer:

a) The odds for \(E\) are \(1\) to \(9\)
b) The odds against \(E\) are \(9\) to \(1\)