Question

PLEASE HELP

4. The table shows the probabilities of a response chocolate or vanilla when asking a child or adult. Use the formula for conditional probability to determine independence.

Chocolate | Vanilla | Total
Adults 0.21 0.39 0.60

Children 0.14 0.26 0.40

Total 0.35 0.65 1.00


a. Are the events “Chocolate” and “Adults” independent? Why or why not?
b. Are the events “Children” and “Chocolate” independent? Why or why not?
c. Are the events “Vanilla” and “Children” independent? Why or why not?

Answer 1

All events are independent

Explanation:

You are given the table

`\begin{array}{cccc}&\text{Chocolate}&\text{Vanilla}&\text{Total}\\\text{Adults}&0.21&0.39&0.60\\\text{Children}&0.14&0.26&0.40\\\text{Total}&0.35&0.65&1.00\end{array}`

Two events A and B are independent when

`Pr(A\cap B)=Pr(A)\cdot Pr(B)`

a) A="Chocolate"

B="Adults"

A and B="Chocolate and Adults"

`Pr(A)=0.35\\ \\Pr(B)=0.60\\ \\Pr(A\cap B)=0.21`

Since
`0.35\cdot 0.60=0.21` events are independent

b) A="Children"

B="Chocolate"

A and B="Children and Chocolate"

`Pr(A)=0.40\\ \\Pr(B)=0.35\\ \\Pr(A\cap B)=0.14`

Since
`0.40\cdot 0.35=0.14` events are independent

c) A="Vanilla"

B="Children"

A and B="Vanilla and Children"

`Pr(A)=0.65\\ \\Pr(B)=0.40\\ \\Pr(A\cap B)=0.26`

Since
`0.65\cdot 0.40=0.26` events are independent