Questions: Math 131-Practice Midterm Exam 2 SOLUTIONS Fall 2024 1. Assume that S is a sample space and A and B are two events. If n(S)=85, n(A ∪ B)=53, n(B)=28, and n(¬A)=35, find the following. Leave any probabilities as unreduced fractions. Show any calculations you are performing, and circle your answer. (a) n(A) n(A)=n(S)-n(¬A)=85-35=50 (b) P(A) P(A)=n(A)/n(S)=50/85 (c) n(A ∩ B) n(A ∩ B)=n(A)+n(B)-n(A ∪ B) n(A ∩ B)=50+28-53=25 (d) P(A ∩ B) P(A ∩ B)=n(A ∩ B)/n(S)=25/85

$Math 131-Practice Midterm Exam 2 SOLUTIONS Fall 2024 1. Assume that S is a sample space and A and B are two events. If n(S)=85, n(A ∪ B)=53, n(B)=28, and n(¬A)=35, find the following. Leave any probabilities as unreduced fractions. Show any calculations you are performing, and circle your answer. (a) n(A) n(A)=n(S)-n(¬A)=85-35=50 (b) P(A) P(A)=n(A)/n(S)=50/85 (c) n(A ∩ B) n(A ∩ B)=n(A)+n(B)-n(A ∪ B) n(A ∩ B)=50+28-53=25 (d) P(A ∩ B) P(A ∩ B)=n(A ∩ B)/n(S)=25/85$

Transcript text: Math 131-Practice Midterm Exam 2 SOLUTIONS Fall 2024 1. Assume that $S$ is a sample space and $A$ and $B$ are two events. If $n(S)=85, n(A \cup B)=53, n(B)=28$, and $n(\bar{A})=35$, find the following. Leave any probabilities as unreduced fractions. Show any calculations you are performing, and circle your answer. (a) $n(A)$ \[ n(A)=n(S)-n(\bar{A})=85-35=50 \] (b) $P(A)$ \[ P(A)=\frac{n(A)}{n(S)}=\frac{50}{85} \] (c) $n(A \cap B)$ \[ n(A \cap B)=n(A)+n(B)-n(A \cup B) \\ n(A \cap B)=50+28-53=25 \] (d) $P(A \cap B)$ \[ P(A \cap B)=\frac{n(A \cap B)}{n(S)}=\frac{25}{85} \]

Solution

Solution Steps

To solve the given problems, we need to use the basic principles of set theory and probability. Here are the steps for each part:

(a) Find $ n(A) $:
- Use the complement rule: $ n(A) = n(S) - n(\bar{A}) $.
(b) Find $ P(A) $:
- Use the definition of probability: $ P(A) = \frac{n(A)}{n(S)} $.
(c) Find $ n(A \cap B) $:
- Use the principle of inclusion-exclusion: $ n(A \cap B) = n(A) + n(B) - n(A \cup B) $.

Step 1: Calculate $ n(A) $

Using the complement rule, we find $ n(A) $ as follows: \[ n(A) = n(S) - n(\bar{A}) = 85 - 35 = 50 \]

Step 2: Calculate $ P(A) $

The probability of event $ A $ is calculated using the formula: \[ P(A) = \frac{n(A)}{n(S)} = \frac{50}{85} \approx 0.5882 \]

Step 3: Calculate $ n(A \cap B) $

To find the number of elements in the intersection of events $ A $ and $ B $, we use the principle of inclusion-exclusion: \[ n(A \cap B) = n(A) + n(B) - n(A \cup B) = 50 + 28 - 53 = 25 \]