Question

A number between 1 and 10, inclusive, is randomly chosen, and the events A and B are defined as follows:

A: [The number is even.]

B: [The number is less than 7.]

a. Identify the sample points in the event $\mathbf{A}\mathbf{\cup }\mathbf{B}$.

b. Identify the sample points in the event $\mathbf{A}\mathbf{\cap }\mathbf{B}$.

c. Which expression represents the event that the number is even or less than 7 or both?

d. Which expression represents the event that the number is both even and less than 7?

Short answer

1. $\mathbf{A}\mathbf{\cup }\mathbf{B}\mathbf{=}\mathbf{[}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{3}\mathbf{,}\mathbf{4}\mathbf{,}\mathbf{5}\mathbf{,}\mathbf{6}\mathbf{]}$
   
2. $\mathbf{A}\mathbf{\cap }\mathbf{B}\mathbf{=}\mathbf{[}\mathbf{2}\mathbf{,}\mathbf{4}\mathbf{,}\mathbf{6}\mathbf{]}$
   
3. $\mathbf{A}\mathbf{\cup }\mathbf{B}$
   
4. $\mathbf{A}\mathbf{\cap }\mathbf{B}$
   

Step-by-step solution

Determine the sample points in the event

A sample point is one of the experiment's possible outcomes in a probabilistic experiment. Sample space refers to the collection of all sample points.

So, a number between 1 and 10 is:

[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

Here,

A: [The number is even]

Equivalently,

A: [2, 4, 6, 8, 10]

B: [The number less than 7]

Equivalently,

B: [1, 2, 3, 4, 5, 6]

Union of sets A and B $\mathbf{A}\mathbf{\cup }\mathbf{B}$:

It identifies a set that occurs in either set A or B or both sets. We can observe that elements 1, 2, 3, 4, 5, and 6 occur in either set A or B.

Therefore, $\mathbf{A}\mathbf{\cup }\mathbf{B}\mathbf{=}\mathbf{[}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{3}\mathbf{,}\mathbf{4}\mathbf{,}\mathbf{5}\mathbf{,}\mathbf{6}\mathbf{]}$

Determine the sample points in the event

Intersect of set A and B, $\mathbf{A}\mathbf{\cap }\mathbf{B}$

It identifies a set that is common in set A and set B. We can observe that elements 1, 2, 3, 4, 5, and 6 are common in sets A and B.

Therefore, $\mathbf{A}\mathbf{\cap }\mathbf{B}\mathbf{=}\mathbf{[}\mathbf{2}\mathbf{,}\mathbf{4}\mathbf{,}\mathbf{6}\mathbf{]}$

Hence,2, 4, 6 are the sample points.

Identify the expression which represents the event that the number is even or less than 7 or both

The event which represents the number is even, less than 7, or both are:

$\mathbf{A}\mathbf{\cup }\mathbf{B}\mathbf{=}\mathbf{[}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{3}\mathbf{,}\mathbf{4}\mathbf{,}\mathbf{5}\mathbf{,}\mathbf{6}\mathbf{]}$

Because this includes even, less than 7, or both the numbers, hence,$\mathbf{A}\mathbf{\cup }\mathbf{B}$represents the expression.

Identify the expression which represents the event that the number is both even and less than 7

The event which represents the number is both even and less than seven is:

$\mathbf{A}\mathbf{\cap }\mathbf{B}\mathbf{=}\mathbf{[}\mathbf{2}\mathbf{,}\mathbf{4}\mathbf{,}\mathbf{6}\mathbf{]}$

Because this includes both even and less than seven numbers,

Hence, $\mathbf{A}\mathbf{\cap }\mathbf{B}$represents the expression.