QuestionFind the number of employees in the Venn diagram for Events A (health insurance) and B (retirement plan) with $P(A|B) = \frac{25}{28}$ , 35 with A, and 28 with B.

Studdy Solution

STEP 1

Assumptions1. Event A represents the health insurance benefit. . Event B represents the retirement savings plan.
3. The probability that an employee receives health insurance benefits, given that they also participate in a retirement savings plan is $\frac{25}{28}$ .
4. There are35 employees who receive health insurance benefits.
5. There are28 employees who participate in a retirement savings plan.

STEP 2

First, we need to find the number of employees who both receive health insurance benefits and participate in a retirement savings plan. This can be found by multiplying the probability that an employee receives health insurance benefits, given that they also participate in a retirement savings plan by the total number of employees who participate in a retirement savings plan.
$Number\, of\, employees\, in\, both\, A\, and\, B = Probability(A|B) \\times Number\, of\, employees\, in\, B$

STEP 3

Now, plug in the given values for the probability and the number of employees who participate in a retirement savings plan to calculate the number of employees who both receive health insurance benefits and participate in a retirement savings plan.
$Number\, of\, employees\, in\, both\, A\, and\, B = \frac{25}{28} \\times28$

STEP 4

Calculate the number of employees who both receive health insurance benefits and participate in a retirement savings plan.
$Number\, of\, employees\, in\, both\, A\, and\, B = \frac{25}{28} \\times28 =25$

STEP 5

Now that we have the number of employees who both receive health insurance benefits and participate in a retirement savings plan, we can find the number of employees who only receive health insurance benefits. This can be found by subtracting the number of employees who both receive health insurance benefits and participate in a retirement savings plan from the total number of employees who receive health insurance benefits.
$Number\, of\, employees\, in\, A\, only = Number\, of\, employees\, in\, A - Number\, of\, employees\, in\, both\, A\, and\, B$

STEP 6

Plug in the values for the number of employees who receive health insurance benefits and the number of employees who both receive health insurance benefits and participate in a retirement savings plan to calculate the number of employees who only receive health insurance benefits.
$Number\, of\, employees\, in\, A\, only =35 -25$

STEP 7

Calculate the number of employees who only receive health insurance benefits.
$Number\, of\, employees\, in\, A\, only =35 -25 =10$

STEP 8

Similarly, we can find the number of employees who only participate in a retirement savings plan. This can be found by subtracting the number of employees who both receive health insurance benefits and participate in a retirement savings plan from the total number of employees who participate in a retirement savings plan.
$Number\, of\, employees\, in\, B\, only = Number\, of\, employees\, in\, B - Number\, of\, employees\, in\, both\, A\, and\, B$

STEP 9

Plug in the values for the number of employees who participate in a retirement savings plan and the number of employees who both receive health insurance benefits and participate in a retirement savings plan to calculate the number of employees who only participate in a retirement savings plan.
$Number\, of\, employees\, in\, B\, only =28 -25$

STEP 10

Calculate the number of employees who only participate in a retirement savings plan.
$Number\, of\, employees\, in\, B\, only =28 -25 =3$ So, the Venn diagram would be filled as follows-10 employees only receive health insurance benefits (Event A only) -3 employees only participate in a retirement savings plan (Event B only) -25 employees both receive health insurance benefits and participate in a retirement savings plan (Both Event A and B)

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