Complementary events, addition and multiplication
Answers are for parents and teachers.
1
In a class of 28, 15 study German, 10 study French and 4 study both. What is the probability that a randomly chosen student studies neither?
At least one language is studied by 15 + 10 − 4 = 21 students; neither by 7: 7/28 = 1/4.
2
The probability of success in one attempt is 0.7. Find the probability of at least one success in three independent attempts.
1 − 0.3³ = 1 − 0.027 = 0.973.
3
At a factory 10 % of the items have a colour defect, 5 % a size defect and 2 % both. Find the probability that a randomly chosen item has no defect. Are the two defects independent events?
At least one defect: 0.1 + 0.05 − 0.02 = 0.13, so no defect: 1 − 0.13 = 0.87. Since 0.1 · 0.05 = 0.005 ≠ 0.02, the events are dependent.
4
Two numbers from 1 to 10 are chosen without replacement. What is the probability that both are even?
5/10 · 4/9 = 20/90 = 2/9.
5
Why is it convenient to find the probability of “at least one” as 1 − P(none)?
“At least one” contains many cases (1, 2, …), whereas “none” is a single case. Computing the complementary event and subtracting from 1 is much shorter.