An insurance company divides its customers into different risk groups. Suppose that 60% of the customers are in group A (low risk), 30% are in group B (medium risk) and 10% are in group C (high risk).
The probabilities of loss occurence for customers in the different groups are as follows:
- Group A: 0.1
- Group B: 0.25
- Group C: 0.74
Calculate the probability that a loss occurs to a randomly chosen customer of the insurance company.
So first, lets start by noting what we already have (L – occurrence of loss):
P(A) = 0.6 given
P(B) = 0.3 given
P(C) = 0.1 given
P(L|A) = 0.1 given => P(L$^c$|A) = 1- 0.1 = 0.9
P(L|B) = 0.25 given => P(L$^c$|B) = 1 – 0.25 = 0.75
P(L|C) = 0.74 given => P(L$^c$|C) = 1 – 0.74 = 0.26
I'm not sure exactly whether the rule of total probability should be applied here to find the P(L) and if the rule of total probability is needed here, I'm rather confused as to how to calculate it with 3 different events? (A,B & C).
Hence, what I tried is:
P(L) = 1 – P(L$^c$) – event that no loss occurs for anyone
P(L$^c$) = P(L$^c$|A)*P(A)*P(L$^c$|B)*P(B)*P(L$^c$|C)*P(C) = 0.9*0.6*0.75*0.3*0.26*0.1 = 0.003159 the chance that no loss occurs to anyone
From here it follows that:
P(L) = 1 – 0.003159 = 0.996841 chance that loss occurs to a randomly chcosen customer of the company
So am I completely wrong or did I correctly solve this? I don't have a solution with which to compare it, hence, I'm asking you, so thank you in advance for your time! Any insights are much appreciated!
Best Answer
You can't multiply all the events like that to calculate $P(L')$. We start with first calculating the probability that he lies in the A loss group or the B loss group or the C loss group.
P(customer is in A loss group) = P(customer belongs to A) $\cdot$ P(an A customer is in loss)
$$P_{A} = 0.6\cdot 0.1 = 0.06$$
P(customer is in B loss group) = P(customer belongs to B) $\cdot$ P(a B customer is in loss)
$$P_{B} = 0.25\cdot 0.3 = 0.075$$
P(customer is in C loss group) = P(customer belongs to C) $\cdot$ P(a C customer is in loss)
$$P_{C} = 0.74\cdot 0.1 = 0.074$$
Since the customer can be in any loss group, we sum it all up to get
$$P = P_{A}+P_{B}+P_{C} = 0.06+0.075+0.074 = 0.209$$