Key idea
For “at least one success” problems, it is usually easier to use the complement: first find the chance of no successes, then subtract from 1.
Method
- Identify the success probability for one trial, and the number of trials if repeated trials are involved.
- Compute the probability of no success in a single trial: if success is (p), then no success is (1-p).
- Extend to all trials if they are independent: multiply the no-success probability across all trials.
- For (n) independent trials, this is ((1-p)^n).
- Subtract from 1:
[
P(\text{at least one success}) = 1 - P(\text{no successes}).
]
- Simplify the final answer as requested.
Check
Your answer should be between 0 and 1. Also, if the success chance increases or the number of trials increases, the probability of at least one success should not get smaller.