The number of balls we get from our grandpa follows the poisson distribution with $\lambda =1/5$ ball/year. The probability of getting a red ball from him is 0.08 .
Q: Calculate the probability of getting a red ball in the upcoming 5 years.
I know that i have to use the total probability theorem but i cant figure out how. Any help is appreciated.
Let $N$ be the number of balls obtained from grandpa and let $R$ be the number of red balls with $0 \le R \le N$. Clearly, if you know $N$, then $R \sim \mathcal{B}(N, p)$ where $p = 0.08$. So you can condition on $N$ to get $$ \mathbb{P}[R = 0] = \sum_{n=0}^\infty \mathbb{P}[R = 0, N = n] = \sum_{n=0}^\infty \mathbb{P}[R = 0 | N = n] \mathbb{P}[N=n] $$ and both factors are now easy to compute. Can you finish?