Question

a 6-sided die with faces labeled 1 to 6 will be rolled once.
the 6 possible outcomes are listed below.
note that each outcome has the same probability.
complete parts (a) through (c). write the probabilities as fractions.

(a) check the outcomes for each event below. then, enter the probability of the event.

table with columns: outcomes (1, 2, 3, 4, 5, 6) and probability; rows: event a: rolling an odd number, event b: rolling a number less than 5, event a or b: rolling an odd number or rolling a number less than 5, event a and b: rolling an odd number and rolling a number less than 5

(b) compute the following.
\\( p(a) + p(b) - p(a \text{ and } b) = \square \\)

(c) select the answer that makes the equation true.
\\( p(a) + p(b) - p(a \text{ and } b) = \text{select} \\)

Explanation:

Part (a)

Event A: Rolling an odd number

Step1: Identify odd outcomes

Odd numbers on a die (1 - 6) are 1, 3, 5. So there are 3 favorable outcomes.
Total outcomes = 6.

Step2: Calculate probability

Probability \( P(A)=\frac{\text{Number of favorable outcomes}}{\text{Total outcomes}}=\frac{3}{6}=\frac{1}{2} \)

Event B: Rolling a number less than 5

Step1: Identify numbers < 5

Numbers less than 5 on a die are 1, 2, 3, 4. So 4 favorable outcomes.

Step2: Calculate probability

\( P(B)=\frac{4}{6}=\frac{2}{3} \)

Event A or B: Rolling an odd number or a number less than 5

Step1: Identify favorable outcomes

Outcomes in A or B: 1, 2, 3, 4, 5 (since 1,3,5 are odd; 2,4 are <5 and not odd). So 5 favorable outcomes.

Step2: Calculate probability

\( P(A \text{ or } B)=\frac{5}{6} \)

Event A and B: Rolling an odd number and a number less than 5

Step1: Identify common outcomes

Outcomes in both A and B: 1, 3 (odd and <5). So 2 favorable outcomes.

Step2: Calculate probability

\( P(A \text{ and } B)=\frac{2}{6}=\frac{1}{3} \)

Part (b)

Step1: Substitute values

We know \( P(A)=\frac{1}{2} \), \( P(B)=\frac{2}{3} \), \( P(A \text{ and } B)=\frac{1}{3} \)
So \( P(A)+P(B)-P(A \text{ and } B)=\frac{1}{2}+\frac{2}{3}-\frac{1}{3} \)

Step2: Simplify

First, \( \frac{2}{3}-\frac{1}{3}=\frac{1}{3} \)
Then, \( \frac{1}{2}+\frac{1}{3}=\frac{3 + 2}{6}=\frac{5}{6} \)

Part (c)

From part (b), we found \( P(A)+P(B)-P(A \text{ and } B)=\frac{5}{6} \), and from part (a), \( P(A \text{ or } B)=\frac{5}{6} \). So the equation \( P(A)+P(B)-P(A \text{ and } B)=P(A \text{ or } B) \) holds.

Final Answers

(a)

• Event A: Probability \( \boldsymbol{\frac{1}{2}} \) (Outcomes: 1, 3, 5 checked)
• Event B: Probability \( \boldsymbol{\frac{2}{3}} \) (Outcomes: 1, 2, 3, 4 checked)
• Event A or B: Probability \( \boldsymbol{\frac{5}{6}} \) (Outcomes: 1, 2, 3, 4, 5 checked)
• Event A and B: Probability \( \boldsymbol{\frac{1}{3}} \) (Outcomes: 1, 3 checked)

(b)

\( P(A)+P(B)-P(A \text{ and } B)=\boldsymbol{\frac{5}{6}} \)

(c)

\( P(A)+P(B)-P(A \text{ and } B)=\boldsymbol{P(A \text{ or } B)} \) (with \( P(A \text{ or } B)=\frac{5}{6} \))

Answer:

Part (a)

Event A: Rolling an odd number

Step1: Identify odd outcomes

Odd numbers on a die (1 - 6) are 1, 3, 5. So there are 3 favorable outcomes.
Total outcomes = 6.

Step2: Calculate probability

Probability \( P(A)=\frac{\text{Number of favorable outcomes}}{\text{Total outcomes}}=\frac{3}{6}=\frac{1}{2} \)

Event B: Rolling a number less than 5

Step1: Identify numbers < 5

Numbers less than 5 on a die are 1, 2, 3, 4. So 4 favorable outcomes.

Step2: Calculate probability

\( P(B)=\frac{4}{6}=\frac{2}{3} \)

Event A or B: Rolling an odd number or a number less than 5

Step1: Identify favorable outcomes

Outcomes in A or B: 1, 2, 3, 4, 5 (since 1,3,5 are odd; 2,4 are <5 and not odd). So 5 favorable outcomes.

Step2: Calculate probability

\( P(A \text{ or } B)=\frac{5}{6} \)

Event A and B: Rolling an odd number and a number less than 5

Step1: Identify common outcomes

Outcomes in both A and B: 1, 3 (odd and <5). So 2 favorable outcomes.

Step2: Calculate probability

\( P(A \text{ and } B)=\frac{2}{6}=\frac{1}{3} \)

Part (b)

Step1: Substitute values

We know \( P(A)=\frac{1}{2} \), \( P(B)=\frac{2}{3} \), \( P(A \text{ and } B)=\frac{1}{3} \)
So \( P(A)+P(B)-P(A \text{ and } B)=\frac{1}{2}+\frac{2}{3}-\frac{1}{3} \)

Step2: Simplify

First, \( \frac{2}{3}-\frac{1}{3}=\frac{1}{3} \)
Then, \( \frac{1}{2}+\frac{1}{3}=\frac{3 + 2}{6}=\frac{5}{6} \)

Part (c)

From part (b), we found \( P(A)+P(B)-P(A \text{ and } B)=\frac{5}{6} \), and from part (a), \( P(A \text{ or } B)=\frac{5}{6} \). So the equation \( P(A)+P(B)-P(A \text{ and } B)=P(A \text{ or } B) \) holds.

Final Answers

(a)

• Event A: Probability \( \boldsymbol{\frac{1}{2}} \) (Outcomes: 1, 3, 5 checked)
• Event B: Probability \( \boldsymbol{\frac{2}{3}} \) (Outcomes: 1, 2, 3, 4 checked)
• Event A or B: Probability \( \boldsymbol{\frac{5}{6}} \) (Outcomes: 1, 2, 3, 4, 5 checked)
• Event A and B: Probability \( \boldsymbol{\frac{1}{3}} \) (Outcomes: 1, 3 checked)

(b)

\( P(A)+P(B)-P(A \text{ and } B)=\boldsymbol{\frac{5}{6}} \)

(c)

\( P(A)+P(B)-P(A \text{ and } B)=\boldsymbol{P(A \text{ or } B)} \) (with \( P(A \text{ or } B)=\frac{5}{6} \))