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Multistage probabilities
Complementary, independent, dependent, incompatible events and probability trees.
Questions: 10Estimated time: 19 minutes
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Question 1out of 10
An experiment is repeated independently three times. The probability of event A on each trial is 80%. Find the probability that A occurs on all three trials, as a decimal.
Multiply probabilities for independent events occurring together. Convert percentages to decimals first: P(A and B) = (a/100)(b/100). Three independent repeated occurrences have probability (p/100)³.
Question 2out of 10
Independent events A and B have probabilities 50% and 20%. Find P(A and B) as a decimal.
Multiply probabilities for independent events occurring together. Convert percentages to decimals first: P(A and B) = (a/100)(b/100). Three independent repeated occurrences have probability (p/100)³.
Question 3out of 10
A box has 4 red and 3 blue balls. Two are drawn without replacement. Find the probability both are red.
Without replacement, both-red probability is r/(r + b) × (r − 1)/(r + b − 1). With replacement the composition stays unchanged, giving [r/(r + b)]². Check whether the first ball is replaced.
Question 4out of 10
In each of 3 independent trials, event A occurs with probability 50%. What is the probability that A occurs at least once?
Complement probability is 1 − P(A), or 100% − p%. An event and its complement cover all outcomes and do not overlap.
Question 5out of 10
How are “and” and “or” interpreted in probability?
In a probability tree, multiply successive probabilities along a path, using the appropriate conditional probabilities. Add probabilities of mutually exclusive favourable paths. For disjoint A and B with equally likely outcomes, P(A or B) = (a + b)/n.
Question 6out of 10
An experiment is repeated independently three times. The probability of event A on each trial is 80%. Find the probability that A occurs on all three trials, as a decimal.
Multiply probabilities for independent events occurring together. Convert percentages to decimals first: P(A and B) = (a/100)(b/100). Three independent repeated occurrences have probability (p/100)³.
Question 7out of 10
How is an event probability calculated in a probability tree?
In a probability tree, multiply successive probabilities along a path, using the appropriate conditional probabilities. Add probabilities of mutually exclusive favourable paths. For disjoint A and B with equally likely outcomes, P(A or B) = (a + b)/n.
Question 8out of 10
There are 10 equally likely outcomes. Event A has 2 favourable outcomes and event B has 3; A and B are mutually exclusive. Calculate the probability that either A or B occurs, i.e. P(A ∪ B).
In a probability tree, multiply successive probabilities along a path, using the appropriate conditional probabilities. Add probabilities of mutually exclusive favourable paths. For disjoint A and B with equally likely outcomes, P(A or B) = (a + b)/n.
Question 9out of 10
There are 12 equally likely outcomes. Event A has 3 favourable outcomes and event B has 3; A and B are mutually exclusive. Calculate the probability that either A or B occurs, i.e. P(A ∪ B).
In a probability tree, multiply successive probabilities along a path, using the appropriate conditional probabilities. Add probabilities of mutually exclusive favourable paths. For disjoint A and B with equally likely outcomes, P(A or B) = (a + b)/n.
Question 10out of 10
A box has 5 red and 5 blue balls. Two are drawn with replacement. Find the probability both are red.
Without replacement, both-red probability is r/(r + b) × (r − 1)/(r + b − 1). With replacement the composition stays unchanged, giving [r/(r + b)]². Check whether the first ball is replaced.
Time left: 00:00:00
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Sample questions
- Event A has probability ___%. Find its complement probability.
- Independent events A and B have probabilities ___% and ___%. Find P(A and B) as a decimal.
- A box has ___ red and ___ blue balls. Two are drawn without replacement. Find the probability both are red.
- A box has ___ red and ___ blue balls. Two are drawn with replacement. Find the probability both are red.
- Of ___ equally likely outcomes, ___ favour A and ___ favour B; the events are incompatible. Find P(A or B).
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