Probability Question (Ambiguous)

#1
SOLVED BY MYSELF, YOU CAN CLOSE THE THREAD :)

Hi guys,

I'm new to this forum, and some of you might find it annoying that I post a question straight off, but the assignment is due tomorrow and I have two different answers. I do promise to keep posting in threads (when I have something to say) :)

It's basic probability. Here goes:

Insurance company X sells a number of different policies, among which 60% are for drivers, 40% for homeowners and 20% for both types of policies. Let A1 be people with only a motor policy, A2 those with only a homeowner policy, A3 people with both types and A4 those with other types of policies.

(i) Assume a person is selected at random. What is the probability that this person belongs to A1/A2/A3/A4 ? (basically they want us to get probability of each separately).

(ii) Let B denote the event that a policyholder will renew at least one of the car or home insurance policies. Based on past experience, we can assume that

P(B|A1) = 0.6,
P(B|A2)= 0.7,
P(B|A3)= 0.8

Given that the person selected at random has a car or a home insurance policy, what is the probability that this person will renew at least one of those policies?

(It was I who made some bits italic and bold).


My Solution

(i) Basic Venn diagram, get
P(A1)=0.4
P(A2)=0.2
P(A3)=0.2
P(A4)=0.2

(ii) First of all, I assume that you're given that the person is allowed to have both policies (A3), because it says "at least one".

From the given information, I worked out
P(B ∩ A1)=0.24
P(B ∩ A2)=0.14
P(B ∩ A3)=0.16

Solution 1

I drew a tree diagram (numbers in brackets represent probabilities)

  • A1 (0.4) ----> B (0.6) so total prob.= 0.24
  • A2 (0.2) ----> B(0.7) so total prob. = 0.14
  • A3 (0.2) ----> B(0.8) so total prob. = 0.16

Then I just added the "total" probabilities to get 0.54

Solution 2

I said that I'm looking for P(B | (A1 U A2 U A3))

which equals to P(B ∩ (A1 U A2 U A3)) / P(A1 U A2 U A3)

which equals to (0.24 + 0.16 + 0.14) / (0.4 + 0.2 + 0.2)

which is 0.675

Which one is right? And what's the difference between the two approaches?

Thanks!
 
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