Probability of engine failure A 2-engine airplane has 0.23 probability of an engine failure in general, 0.1 is probability of a failure in the right engine and 0.15 is the probability of a failure in the left engine.


What is the probability of a failure in both engine?
What is the probability of a failure in only one engine?


I solved it as follows:


$P=0.1\cdot 0.15$
$P=0.1\cdot(1-0.15)+0.15 \cdot(1-0.1)$


I didn’t develop intuition for probability yet and the fact that I didn’t use the probability of failure in general makes me think I got it all wrong.
Your solution would be OK if the events involved were independent. But they aren't.

Call $A=$ failure in 1st engine and $B=$failure in second engine.

You must use this:
$$P(A)+P(B)=P(A\cup B)+P(A\cap B)$$

That is
$$0.1+0.15=0.23+P(A\cap B)$$
so the probability of a double failure is $0.02$.
In general, the probability of a single failure of an engine is p.  The probability that one will fail on a twin-engine aircraft is 2p.  The probability that both will fail is p^2.  There are similar relationships for more engines.  2p^3, p^4, etc.

In this specific case, you have 0.1 in the right side and 0.15 on the left side, so you have 0.1 + 0.15 = 0.25 as the probability of either failing.  Not sure where your 0.23 came from.  For both to fail, you would get 0.1 x 0.15 = 0.015.