Question

Records show that the probability is 0.00006 that a car will have a flat tire while driving through a certain tunnel. Find the probability that at least 2 of 10,000 cars passing through this tunnel will have flat tires.

Answer 1

This question can be solved with help of Poisson's distribution .
According to Poisson's distribution ,
If n quantity is large and p quantity is small then,
f(x) = $\bold{e^{-np}\frac{(np)^x}{x!}}}$
Here x = 0, 1, 2, 3, .......

A/C to we have to find probability at least two cars .
so, probability of at least two cars = 1 - probability of one cars - probability of no car
e.g., P(x ≥ 2) = 1 - [P(x = 1) + P(x = 0) ]
= 1 - $\bold{e^{-np}\frac{(np)^1}{1!}}}$ - $\bold{e^{-np}\frac{(np)^0}{0!}}}$
= 1 - $\bold{e^{-np}\frac{np}{1}}$ - $\bold{e^{-np}\frac{1}{1}}$

Now, n = 10000 and p = 0.00006
np = 10000 × 0.00006 = 0.6

∴ P(x ≥ 2) = 1 - $\bold{e^{-0.6}0.6}$ - $\bold{e^{-0.6}}$
= 1 - $\bold{1.6e^{-0.6}}$
= 0.1219

Hence , probability = 0.1219