Question

An insurance company receives 25 claims in a certain month. Company policy mandates that a random selection of 5 of these claims be thoroughly investigated.if two of the 25 claims are fraudulent, find the probability that
(I)both claims are through investigated
(II) neither fraudulent claim is investigated

Answer 1

(I)

Solution:

We choose 5 claims to investigate: 3 cleared and 2 fraudulent. First we choose from some amount out of 25, then from some amount out of 24 and so on, so the denominator of probability is $(25\cdot24\cdot23\cdot22\cdot21)$. When we choose 3 cleared claimes, it happenes with probability 23 out of all claimes, then 22 out of rest claimes then 21 out of rest claimes. When we choose 2 fraudulent claimes, it happenes with probability 2 out of rest claimes, then 1 out of rest claimes. So the numerator of probability is $(23\cdot22\cdot21)(2\cdot1)$. But it is the probability of just one sequence of claimes: Cleared claim, cleared claim, cleared claim, fraudulent claim, fraudulent claim. There are $C_5^2$ of such possible sequences. The order of claims doesn't affect on numerator and denominator.

So the probability is $p=\frac{(23\cdot22\cdot21)(2\cdot1)}{(25\cdot24\cdot23\cdot22\cdot21)}\cdot C_{5}^{2} = \frac{1}{30} \approx 0.033$

Answer: 0.033

(II)

Solution:

Same logic as in previous one. There are $C_5^0 = 1$ of such possible sequences of 5 cleared claimes. first we choose 23 out of 25 cleared claimes, then 22 out of 24 cleared claims, and so on.

Probability is $p=\frac{23\cdot22\cdot21\cdot20\cdot19}{25\cdot24\cdot23\cdot22\cdot21} \cdot C_5^0 = \frac{19}{30} \approx 0.633$

Answer: 0.633