Question

Say you test drive 3 different cars. Let A1 = You like car 1; A2 = You like car 2; and A3 = You like car 3. Suppose P(A1) = .5; P(A2) = .6; P(A3)=.7 and P(A1 or A2) = .8 and P(A2 and A3) = .4 and P(A1 or A2 or A3) = .9 d. Are A2 and A3 independent events? Make sure you can show your work for this in two ways.

Answer 1

`A_2, A_3` are not independent events.

Explanation:

We are given the following in the question:

`A_1:\text{You like car 1}\\A_2:\text{You like car 2}\\A_3:\text{You like car 3}\\P(A_1) = 0.5\\P(A_2) = 0.6\\P(A_3) = 0.7\\P(A_1\cup A_2) = 0.8\\P(A_2\cap A_3) = 0.4\\P(A_1\cup A_2\cup A_3) = 0.9`

Independent events:

• Two events A and B is said to be independent if

`P(A\cap B) = P(A)* P(B)`

Since,

`P(A_2\cap A_3) \\eq P(A_2)* P(A_3)\\0.4\\eq 0.6* 0.7 \\0.4\\eq 0.42`

Thus, they are not independent events.

Now, we evaluate

`P(A_2|A_3) = (P(A_2\cap A_3))/(P(A_3)) = (0.4)/(0.7) = 0.57\\\\P(A_2|A_3) \\eq P(A_2) = 0.6`

Thus, they are not independent event.