James Bond is recording. On a break he crosses an avenue. The cars pass according to a homogeneous Poisson Process with intensity $\lambda = 6$ per minute. He takes $s$ seconds to cross and can dodge (only) one car during his crossing. If a car passes that point and you don't dodge it, then you will get injured. What is the probability that Bond will be injured? (for $s = 10, 20$).
I have problems doing this exercise, honestly I don't know how to do it, the fact of working with minutes and seconds causes me a lot of conflict, can you help me?
My idea was to say that since the cars pass according to a poisson process of intensity $\lambda=6$ per minute then, the cars pass according to a poisson process of intersity $\lambda=\frac{6}{60} =0.1$, then the probability we are looking for is $P[N(t)\geq 2]$, In this case $P[N(10)\geq 2]$ and $P[N(20)\geq 2]$, is this correct?
Your reasoning is correct. Let $X_s$ be the number of cars that arrive in $(0,s)$ - this has Poisson distribution with parameter $\frac {\lambda s}{60}$ (with $\lambda = 6$, this reduces to $\frac s{10}$). The desired probability is then $$ \mathbb P(X_s\geqslant 2) = 1-(\mathbb P(X_s=0)+\mathbb P(X_s=1))\\ = 1 - e^{-\frac s{10}}\left(1 + \frac s{10}\right)\\ $$ For $s=10$ this is $1-2e^{-1}\approx0.264241$ and for $s=20$ this is $1-3e^{-2}\approx0.593994$.