[Math] If you learn that the individual did not like vehicle $\#1$, what now is the probability that he/she liked at least one of the other two vehicles

probability

Consider randomly selecting a single individual and having that person test
drive 3 different vehicles.

Define events $A_1, A_2, A_3$ by:

  • $A_1$ = likes vehicle #1
  • $A_2$ = likes vehicle #2
  • $A_3$ = likes vehicle #3

Suppose that $P(A_1) = .55 \,, P(A_2) = .65 \,, P(A_3) = .7 \,, P(A_1 \cup A_2) = .8 \,, P(A2 \cap A3) = .4, \,, P(A1 \cup A2 \cup A3) = .88$

(a) What is the probability that the individual likes both vehicle #1 and vehicle 2?

(b) Determine and interpret $P(A_2 \mid A_3)$

(c) Are $A_2$ and $A_3$ independent events? Answer in two different way.

(d) If you learn that the individual did not like vehicle #1, what now is the
probability that he/she liked at least one of the other two vehicles?

I am stuck on the fourth part. What I have done so far : I am thinking $P(A_2 \mid A_1') + P(A_3 \mid A_1') + P(A_3 \cap A_2 \mid A_1')$

Best Answer

I am stuck on the fourth part. What I have done so far : I am thinking P(A2 | A1') + P(A3 | A1') + P(A3 intersect A2 | A1')

Close. You want: $\quad\mathsf P(A_2\cup A_3\mid A_1') = \mathsf P(A_2\mid A_1') + \mathsf P(A_3\mid A_1') - \mathsf P(A_2\cap A_3 \mid A_1')$

Can you find these terms ?

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