Vehicles cross a bridge following a Poisson Process with rate of a vehicle per minute. If 5% of these are vans, then:
a) What is the probability that at least 1 van cross the bridge in one hour?
b) Knowing that 10 vans crossed the bridge in one hour, what is the expected number of vehicles that crossed the bridge in this time?
I'm reasoning as follows:
a) Vans $V \sim \mathrm{Pois}(0.5\times60)$, then $\mathbb{P}\{V>0\} = 1 - \mathbb{P}\{V=0\} = 1 - e^{-0.5\times60}$
b) Not Vans $\overline{V} \sim \mathrm{Pois}(0.95 \times 60)$, then $\mathbb{E}[\text{total vehicles}] = 10 + \mathbb{E}[\overline{V}] = 0.95\times 60$
I'd appreciate your help!
Editing with the help of André Nicolas and SiongthyeGoh
a) Vans $V \sim \mathrm{Pois}(0.05\times60)$, then $\mathbb{P}\{V>0\} = 1 - \mathbb{P}\{V=0\} = 1 - e^{-0.05\times60}$
b) Not Vans $\overline{V} \sim \mathrm{Pois}(0.95 \times 60)$, then $\mathbb{E}[\text{total vehicles}] = 10 + \mathbb{E}[\overline{V}] = 10+ 0.95\times 60$
Seems fine. Just some careless mistakes.
For part (a), $5\%$ should be 0.05 rather than 0.5.
For part (b), @Gaffney's comment is right. A term is missing.