Question

question 5
1 pts
suppose events e and f are independent, with p(f)=0.64 and p(e∩f)=0.16. 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: Find \(P(E)\) using the formula for independent events

For independent events \(E\) and \(F\), \(P(E\cap F)=P(E)\times P(F)\). Given \(P(F) = 0.64\) and \(P(E\cap F)=0.16\), we can solve for \(P(E)\) as follows:

$$P(E)=\frac{P(E\cap F)}{P(F)}=\frac{0.16}{0.64}=\frac{1}{4} = 0.25$$

Step2: Calculate the odds for \(E\)

The formula for odds for an event \(E\) is \(\text{Odds for }E=\frac{P(E)}{1 - P(E)}\). Substituting \(P(E)=0.25\), we get \(1 - P(E)=1- 0.25 = 0.75\). Then \(\frac{P(E)}{1 - P(E)}=\frac{0.25}{0.75}=\frac{1}{3}\), so the odds for \(E\) are \(1\) to \(3\).

Step3: Calculate the odds against \(E\)

The formula for odds against an event \(E\) is \(\text{Odds against }E=\frac{1 - P(E)}{P(E)}\). Substituting \(P(E)=0.25\) and \(1 - P(E)=0.75\), we get \(\frac{1 - P(E)}{P(E)}=\frac{0.75}{0.25}=3\), so the odds against \(E\) are \(3\) to \(1\).

Answer:

a) \(1\) to \(3\)
b) \(3\) to \(1\)