Question

The number of injury claims per month is modeled by a random variable 𝑁 with Pr(𝑁 = 𝑛) = 1/(𝑛+1)(𝑛+2), for 𝑛 = 0, 1, 2, 3, · · · . Determine the probability of at least one claim during a particular month, given that there have been at most four claims during the months

Answer 1

We need to compute $P(N ≥ 1|N ≤ 4),$  which is the same as $\dfrac{P(1\leq N\leq 4)}{P(N\leq 4)}$

$$P(1\leq N\leq 4)=P(N=1)+P(N=2)$$

$$+P(N=3)+P(N=4)$$

$$=\dfrac{1}{(1+1)(1+2)}+\dfrac{1}{(2+1)(2+2)}$$

$$+\dfrac{1}{(3+1)(3+2)}+\dfrac{1}{(4+1)(4+2)}$$

$$=\dfrac{1}{6}+\dfrac{1}{12}+\dfrac{1}{20}+\dfrac{1}{30}=\dfrac{1}{3}$$

$$P( N\leq 4)=P(N=0)+P(1\leq N\leq 4)$$

$$=\dfrac{1}{(0+1)(0+2)}+\dfrac{1}{3}=\dfrac{5}{6}$$

Then

$$P(N ≥ 1|N ≤ 4)=\dfrac{P(1\leq N\leq 4)}{P(N\leq 4)}$$

$$=\dfrac{1/3}{5/6}=\dfrac{2}{5}=0.4$$