Question

question 8
1 pts
suppose events e and f are mutually exclusive, with p(f) = 0.60 and p(e ∪ f) = 0.84. 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: Recall formula for mutually exclusive events

For mutually exclusive events \( E \) and \( F \), \( P(E \cup F)=P(E)+P(F) \). We know \( P(F) = 0.60 \) and \( P(E \cup F)=0.84 \), so we can solve for \( P(E) \).
\( P(E)=P(E \cup F)-P(F)=0.84 - 0.60 = 0.24 \)

Step2: Calculate odds for \( E \)

Odds for an event \( E \) is given by \( \frac{P(E)}{1 - P(E)} \). First, find \( 1 - P(E)=1 - 0.24 = 0.76 \). Then, \( \frac{P(E)}{1 - P(E)}=\frac{0.24}{0.76}=\frac{24}{76}=\frac{6}{19}\)? Wait, no, wait, let's simplify correctly. Wait, 0.24 is \( \frac{24}{100}=\frac{6}{25} \), 0.76 is \( \frac{76}{100}=\frac{19}{25} \). Wait, no, wait, odds for \( E \) is \( P(E):(1 - P(E)) \), so \( 0.24:0.76 \). Multiply numerator and denominator by 100 to eliminate decimals: \( 24:76 \), then divide by 4: \( 6:19 \)? Wait, no, wait, maybe I made a mistake. Wait, 0.24 and 0.76. Let's divide both by 0.04: 0.24/0.04 = 6, 0.76/0.04 = 19. Wait, but maybe the problem expects decimal to fraction simplification. Wait, alternatively, 0.24 is 6/25, 0.76 is 19/25, so odds for \( E \) is 6:19? Wait, no, wait, odds for \( E \) is \( P(E) \) to \( P(\text{not } E) \), so \( 0.24 \) to \( 0.76 \). Let's simplify the ratio. Divide both by 0.04: 6 to 19? Wait, but maybe the question wants it as a ratio of integers. Wait, 0.24 is 24/100, 0.76 is 76/100. So 24:76 = 6:19? Wait, but maybe I miscalculated \( P(E) \). Wait, 0.84 - 0.60 is 0.24, that's correct. Then \( 1 - P(E)=0.76 \). So odds for \( E \) is \( P(E):(1 - P(E)) = 0.24:0.76 \). Multiply both by 100: 24:76, divide by 4: 6:19. Wait, but maybe the problem expects decimal to whole numbers. Wait, 0.24 and 0.76, if we multiply by 25: 0.2425 = 6, 0.7625 = 19. So odds for \( E \) is 6 to 19? Wait, but let's check again. Wait, maybe I messed up the formula. Odds for \( E \) is \( \frac{P(E)}{P(\text{not } E)} \), which is \( \frac{0.24}{0.76}=\frac{6}{19}\approx0.3158 \), but as a ratio, it's 6:19.

Step3: Calculate odds against \( E \)

Odds against \( E \) is \( P(\text{not } E):P(E) \), so \( 0.76:0.24 \). Using the same simplification, 0.76:0.24 = 76:24 = 19:6 (dividing by 4).

Wait, but let's re - check the steps:

1. Find \( P(E) \):

Since \( E \) and \( F \) are mutually exclusive, \( P(E\cup F)=P(E)+P(F) \).
Given \( P(E\cup F) = 0.84 \) and \( P(F)=0.60 \), then \( P(E)=0.84 - 0.60=0.24 \).

1. Odds for \( E \):

Odds for \( E=\frac{P(E)}{1 - P(E)}=\frac{0.24}{1 - 0.24}=\frac{0.24}{0.76}=\frac{24}{76}=\frac{6}{19}\) (ratio form: \( 6:19 \))

1. Odds against \( E \):

Odds against \( E=\frac{1 - P(E)}{P(E)}=\frac{0.76}{0.24}=\frac{76}{24}=\frac{19}{6}\) (ratio form: \( 19:6 \))

But wait, the problem says "enter your answer one digit per box". Wait, maybe we made a mistake in the decimal to fraction conversion. Wait, 0.24 is 24/100 = 6/25, 0.76 is 76/100 = 19/25. But maybe the question expects the odds in terms of simplified whole - number ratios. Let's check the decimal values again. 0.24 and 0.76. If we multiply both by 100, we get 24 and 76. Then divide both by 4, we get 6 and 19. So odds for \( E \) is 6 to 19, and odds against \( E \) is 19 to 6.

Answer:

a) The odds for \( E \) are: 6 to 19

b) The odds against \( E \) are: 19 to 6