A plane crashes with probability 0.95 if both of its engines fail. On each flight each engine has a probability of failure of $10^{-5}$. Both engines fail with probability of $10^{-9}$
a) Are the engine failures independent of each other? b) If one engine fails what is the probability that the other one also fails? c) What is the probability that the plane crashes? d) In 10000 flights what is the probability that 1 or more crashes occur?
I think that for a)
If the engine failures are independent $ P(A \cap B) = P(A)P(B) $ So, according to given information
$ 10^{-9} = 10^{-5}*10^{-5} $ this is not correct. so engine failures dependent
for b)
$P(E_2 | E_1) = \frac{P(E_2\cap E_1)}{P(E_1)}=\frac{10^{-9}}{10^{-5}} = 10^{-4}$
for c) i didn't understand what it asks, but i suppose it asks that (C stands for crash) $ P(C) = ? $
for d) Also i didn't understand what it asks for.
what do you think about c and d?
For c). It is more convenient to denote the required probability with $p$ (and not $P(C)$) because you are going to use in d). So $$p=0.95\cdot P(E_1\cap E_2)=0.95\cdot10^{-9}$$
For d). The number $X$ of crashes that occur in $10000$ flights is a binomial random variable with parameters $n=100000$ and $p=0.95\cdot 10^{-9}$ (from c). Thus $$P(X\ge 1)=1-P(X<1)=1-P(X=0)=1-(1-p)^{10000}$$