Probabilities of sequences of independent events

Independent event sequences

Independent trials have no change in the probability of one trial caused by another. Multiply the probability of every required success or failure, and use a complement for an at-least-one event

1. Multiply a specified pattern A pattern that requires an outcome at each trial uses one factor for each required result \(\displaystyle \Pr(S,F,S)=p_1(1-p_2)p_3\)
2. Use the failure complement The probability of failure on a trial is one minus its success probability \(\displaystyle \Pr(E_i^{\prime})=1-p_i\)
3. Use a complement for at least one At least one success is the complement of failure on every trial \(\displaystyle \Pr(\text{at least one})=1-(1-p)^n\)
4. Use each trial's probability When success probabilities differ, use the probability belonging to each trial \(\displaystyle \Pr(S,F)=\frac{2}{3}\times\frac{1}{4}=\frac{1}{6}\)

Worked example

Four independent trials have success probabilities one half, two thirds, three quarters and one fifth. Find the probability of at least one success

1. \(\displaystyle \Pr(E_1^{\prime})=\frac{1}{2},\quad\Pr(E_2^{\prime})=\frac{1}{3},\quad\Pr(E_3^{\prime})=\frac{1}{4},\quad\Pr(E_4^{\prime})=\frac{4}{5}\)Subtract each success probability from 1 to find the corresponding failure probability
2. \(\displaystyle \Pr(\text{no successes})=\frac{1}{2}\times\frac{1}{3}\times\frac{1}{4}\times\frac{4}{5}=\frac{1}{30}\)Independence means that the four failure probabilities are multiplied
3. \(\displaystyle \Pr(\text{at least one success})=1-\Pr(\text{no successes})=1-\frac{1}{30}=\frac{29}{30}\)Subtract the probability of no successes from 1.

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