Are you sure the flight is safe?

The plane will not fly only if all of the engines malfunction. The chances that the engines malfunction are $\frac{1}{2}, \frac{1}{3}, \frac{4}{7}, \frac{1}{8}$ respectively. The chance that all the engines malfunction is $\frac{1}{2} \times \frac{1}{3} \times \frac{4}{7} \times \frac{1}{8} = \frac{1}{84}$ Therefore, the chance that at least one of the engines function is $1- \frac{1}{84} = \frac{83}{84}$ And $83+84=167$

Instead of finding directly to the probability of the airplane to fly safe, we can do it in the other way.

Main idea: Find the probability of the airplane to crash and get 1 subtract to it.

The probability for the engines to malfunction are $\frac{1}{2}$ , $\frac{1}{3}$ , $\frac{4}{7}$ and $\frac{1}{8}$ respectively. So, the probability for all engines to malfunction is $\frac{1}{2}\times\frac{1}{3}\times\frac{4}{7}\times\frac{1}{8}=\frac{1}{64}$

Now, if the airplane does not crash, what does it mean? Yes, it flies!

1 is the overall conditions, so we get $1-\frac{1}{64}=\frac{63}{64}$ and we get 63+64=127

The probability for all engines to fail= 1/2 1/3 4/7*1/8=4/336=1/84 Therefore, the probability at least one engine works ( good enough to fly) is 1-1/84= 83/84. a+b=167

The plane will fly safely if at least one engine works . Therefore, we have to find the probability of at least one engine working.

The probability that at least one engine works is $1 - P(\text{all engines fail})$

Now, probability that all engines fail is

$\left(1 - \dfrac{1}{2} \right) \times \left(1 - \dfrac{2}{3} \right) \times \left(1 - \dfrac{3}{7} \right) \times \left(1 - \dfrac{7}{8} \right)$

Therefore we get the probability of all engines failing as

$\dfrac{1}{2} \times \dfrac{1}{3} \times \dfrac{4}{7} \times \dfrac{1}{8} = \dfrac{1}{84}$

Remember that the probability of at least one engine working was

$P (\text{at least one engine working}) = 1 - P(\text{all engines fail})$

Thus, $P (\text{at least one engine working}) = 1 - \left(\dfrac{1}{84} \right) = \dfrac{83}{84}$ giving us the answer $83 + 84 = \boxed{167}$