A rocket-engine igniter system has n separate small igniters. Any one igniter has a probability $\theta$ of firing independently of the other igniters. Since each is small, the firing of a group of igniters may or may not successfully ignite the propellant. It may be assumed that, if $\mathrm{k}$ igniters out of n fire, the probability that the propellant ignites has one of two forms:
(a) $\alpha \mathrm{k}+\beta \mathrm{k}^{2}$
(b) $1-\gamma^{k}$
where $\alpha, \beta$, and $\gamma$ are known constants. Assuming (a) and
(b) in turn, prove that the probabilities that the propellant ignites are respectively
(a) $n \theta(\alpha+n \theta \beta+\beta-\beta \theta)$
(b) $1-(1-\theta+\gamma \theta)^{\mathrm{n}}$.