A-level · Statistical distributions

Cumulative binomial probabilitiesExam-style questions

Finding P(X ≤ r), P(X < r), P(X ≥ r), P(X > r) and P(a ≤ X ≤ b) for a binomial distribution by building everything from the cumulative function P(X ≤ r) (binomcdf), with care over the off-by-one when an inequality is strict or "at least". Includes the complement for "≥" and "two-stage" problems where the success of one binomial becomes the trial of another.

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11 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 3 Level 4 Level 5

1
Each day, a fishing boat goes out to sea. On any day, the probability that the boat returns with a catch of more than $200$ kg is $0.35$ , independent of all other days.
(a) Find the probability that, in a period of $8$ days, the boat returns with a catch of more than $200$ kg on at least $3$ days.

(3 marks)

A period of $8$ days in which the boat returns with a catch of more than $200$ kg on at least $3$ days is described as a 'good' period.
(b) Find the probability that at least one of $3$ randomly chosen periods of $8$ days is a good period.

(2 marks)

A second fishing boat sails from a different harbour. On any day, the probability that this boat returns with a catch of more than $200$ kg is $0.4$ , independent of all other days.
(c) Use a suitable approximation to find the probability that, in $90$ randomly chosen days, this boat returns with a catch of more than $200$ kg on fewer than $30$ days.

(5 marks)
Level 5 · 10 marksTry it