How do you find the probability of at least one success?
Use the complement rule: P(at least one) = 1 - P(none). Computing the chance of nothing happening is much easier than adding up every way that something happens.
Explain it simply
"At least one" covers one, two, three or more successes, which is a long list. But there is only one way to get none. So work out the probability of none and subtract it from 1.
If events are independent, P(none) is the product of the individual failure probabilities.
Example
You toss a fair coin 3 times. What is the chance of at least one head?
- P(no heads) = 1/2 x 1/2 x 1/2 = 1/8
- P(at least one head) = 1 - 1/8 = 7/8 = 0.875
A second case: a bowler takes a wicket with chance 1/6 each ball, independently. Chance of at least one wicket in 4 balls:
- P(none) = (5/6)^4 = 0.4823
- P(at least one) = 1 - 0.4823 = 0.5177
print(1 - 0.5 ** 3) # 0.875
print(round(1 - (5/6) ** 4, 4)) # 0.5177What to say in the interview
"For at least one, I use 1 minus the probability of none. With independent trials, I multiply the individual no-event probabilities. It also appears in reliability, for example the chance that at least one of several servers is up."
Common mistakes
- Adding probabilities of independent events, such as 4 x 1/6, which can exceed 1 and is wrong.