Question

seven balls numbered 1 to 7 are placed in a bag.
some of the balls are grey and some are white.
the balls numbered 1, 4, and 5 are grey. the balls numbered 2, 3, 6, and 7 are white.
a ball will be selected from the bag at random.
the 7 possible outcomes are listed below.
note that each outcome has the same probability.

image of balls: 1 (grey), 2 (white), 3 (white), 4 (grey), 5 (grey), 6 (white), 7 (white)

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.

chart with outcomes (1,2,3,4,5,6,7) and probability column, and events: event a, event b, event a and b, event a or b

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

Explanation:

Part (a)

Event A: The selected ball has a number from 2 to 5

Step1: Identify outcomes for Event A

Numbers from 2 to 5 are 2, 3, 4, 5. So the outcomes are 2, 3, 4, 5.
Number of favorable outcomes = 4. Total outcomes = 7.

Step2: Calculate probability

Probability of Event A, \( P(A)=\frac{\text{Number of favorable outcomes}}{\text{Total outcomes}}=\frac{4}{7} \)

Event B: The selected ball is grey

Step1: Identify grey balls

Grey balls are 1, 4, 5. So outcomes are 1, 4, 5.
Number of favorable outcomes = 3. Total outcomes = 7.

Step2: Calculate probability

Probability of Event B, \( P(B)=\frac{3}{7} \)

Event A and B: The selected ball has a number from 2 to 5 and is grey

Step1: Identify common outcomes

Numbers from 2 - 5 and grey: 4, 5 (since 1 is not in 2 - 5, 2 and 3 are white). So outcomes are 4, 5.
Number of favorable outcomes = 2. Total outcomes = 7.

Step2: Calculate probability

Probability of \( A \cap B \), \( P(A \cap B)=\frac{2}{7} \)

Event A or B: The selected ball has a number from 2 to 5 or is grey

Step1: Identify union outcomes

Outcomes in A or B: 1, 2, 3, 4, 5 (A: 2,3,4,5; B:1,4,5; union is 1,2,3,4,5). Number of favorable outcomes = 5. Total outcomes = 7.

Step2: Calculate probability

Probability of \( A \cup B \), \( P(A \cup B)=\frac{5}{7} \)

Part (b)

Step1: Use the formula for union of events

The formula for the probability of \( A \cup B \) is \( P(A \cup B)=P(A)+P(B)-P(A \cap B) \). We can also calculate \( P(A)+P(B)-P(A \cap B) \) directly.
We know \( P(A)=\frac{4}{7} \), \( P(B)=\frac{3}{7} \), \( P(A \cap B)=\frac{2}{7} \)

Step2: Substitute values

\( P(A)+P(B)-P(A \cap B)=\frac{4}{7}+\frac{3}{7}-\frac{2}{7}=\frac{4 + 3- 2}{7}=\frac{5}{7} \)

Final Answers

(a)

• Event A: Probability \( \boldsymbol{\frac{4}{7}} \)
• Event B: Probability \( \boldsymbol{\frac{3}{7}} \)
• Event A and B: Probability \( \boldsymbol{\frac{2}{7}} \)
• Event A or B: Probability \( \boldsymbol{\frac{5}{7}} \)

(b)

\( P(A)+P(B)-P(A \cap B)=\boldsymbol{\frac{5}{7}} \)

Answer:

Part (a)

Event A: The selected ball has a number from 2 to 5

Step1: Identify outcomes for Event A

Numbers from 2 to 5 are 2, 3, 4, 5. So the outcomes are 2, 3, 4, 5.
Number of favorable outcomes = 4. Total outcomes = 7.

Step2: Calculate probability

Probability of Event A, \( P(A)=\frac{\text{Number of favorable outcomes}}{\text{Total outcomes}}=\frac{4}{7} \)

Event B: The selected ball is grey

Step1: Identify grey balls

Grey balls are 1, 4, 5. So outcomes are 1, 4, 5.
Number of favorable outcomes = 3. Total outcomes = 7.

Step2: Calculate probability

Probability of Event B, \( P(B)=\frac{3}{7} \)

Event A and B: The selected ball has a number from 2 to 5 and is grey

Step1: Identify common outcomes

Numbers from 2 - 5 and grey: 4, 5 (since 1 is not in 2 - 5, 2 and 3 are white). So outcomes are 4, 5.
Number of favorable outcomes = 2. Total outcomes = 7.

Step2: Calculate probability

Probability of \( A \cap B \), \( P(A \cap B)=\frac{2}{7} \)

Event A or B: The selected ball has a number from 2 to 5 or is grey

Step1: Identify union outcomes

Outcomes in A or B: 1, 2, 3, 4, 5 (A: 2,3,4,5; B:1,4,5; union is 1,2,3,4,5). Number of favorable outcomes = 5. Total outcomes = 7.

Step2: Calculate probability

Probability of \( A \cup B \), \( P(A \cup B)=\frac{5}{7} \)

Part (b)

Step1: Use the formula for union of events

The formula for the probability of \( A \cup B \) is \( P(A \cup B)=P(A)+P(B)-P(A \cap B) \). We can also calculate \( P(A)+P(B)-P(A \cap B) \) directly.
We know \( P(A)=\frac{4}{7} \), \( P(B)=\frac{3}{7} \), \( P(A \cap B)=\frac{2}{7} \)

Step2: Substitute values

\( P(A)+P(B)-P(A \cap B)=\frac{4}{7}+\frac{3}{7}-\frac{2}{7}=\frac{4 + 3- 2}{7}=\frac{5}{7} \)

Final Answers

(a)

• Event A: Probability \( \boldsymbol{\frac{4}{7}} \)
• Event B: Probability \( \boldsymbol{\frac{3}{7}} \)
• Event A and B: Probability \( \boldsymbol{\frac{2}{7}} \)
• Event A or B: Probability \( \boldsymbol{\frac{5}{7}} \)

(b)

\( P(A)+P(B)-P(A \cap B)=\boldsymbol{\frac{5}{7}} \)