Two events are independent if knowing that one event happened does not change the probability of the other event.
Independence means “no update.” If learning B happened leaves the chance of A exactly the same, then the events are independent.
Showing a random 20 of 80 problems.
Example 1
hard
Three independent hypothesis tests, each with significance level α=0.05, are run on null-true data. Find P(at least one false rejection).
Example 2
hard
A,B are independent. P(A∪B)=0.7 and P(B)=0.5. Find P(A).
Example 3
medium
A sensor's two independent readings each fail with probability 0.02. Find P(at least one fails).Sensor 1 → Sensor 2; P(both OK) = 0.98² = 0.9604; P(at least one fails)
Example 4
hard
A coin is flipped 5 times. Find P(exactly 3 heads).
Example 5
easy
A coin and a die are observed as independent trials. Find P(heads and 4 or higher).
Example 6
hard
A simple random sample of 5 voters has P(vote yes)=0.55 for each (with replacement). Find P(at least one no).
Example 7
easy
Two independent diagnostic tests each have P(positive∣healthy)=0.05. Find P(both false positive).
Example 8
easy
A spinner gives red with probability 0.3. Spun twice independently, find P(red on both spins).Spin 1 → Spin 2; highlighted path Red→Red has probability 0.3 × 0.3 = 0.09
Example 9
hard
An A/B test tracks 4 visitors in variant A. Each visitor converts independently with P(convert)=0.4. Find P(exactly 2 of the 4 visitors convert).
Example 10
hard
A fair coin is flipped until heads appears. Find P(exactly 3 flips needed).
Example 11
medium
A weather model gives P(rain today)=0.3 and P(rain tomorrow)=0.4, independent. Find P(rain on at least one day).Today → Tomorrow; P(no rain both) = 0.7 × 0.6 = 0.42; P(rain at least one day)
Example 12
easy
Events A,B are independent. If P(A)=0 what is P(A∩B)?
Example 13
easy
A red die and a blue die are rolled together. Find P(red is 3 and blue is even).Red die → Blue die; highlighted path Red=3→Even has probability 1/6 × 1/2 = 1/12
Example 14
easy
A die is rolled twice. Find P(6 then 6).Roll 1 → Roll 2; highlighted path 6→6
Example 15
medium
Two independent events have P(A)=0.6, P(B)=0.5. Find P(A∪B).
Example 16
hard
Three independent events A,B,C each have probability 0.6. Find P(exactly 2 occur).
Example 17
medium
P(A)=0.4, P(B)=0.5, P(A∩B)=0.25. Are A,B independent?
Example 18
medium
A test for a disease is positive with probability 0.95 if a person is sick. Two independent tests are run on a sick person. Find P(both positive).Test 1 → Test 2; highlighted path Pos→Pos
Example 19
easy
A bag has 5 balls. Draw one WITHOUT replacement, then another. Are the draws independent?
Example 20
medium
Two archers hit a target independently with probabilities 0.7 and 0.8. Find the probability that exactly one hits.Archer A → Archer B; exactly one hit = P(Hit A, Miss B) + P(Miss A, Hit B)