Probability Questions and Answers

A head of school can contact parents by e-mail, letter or cellphone with probabilities 0.4, 0.1 and 0.5 respectively, using only one method. The probability that parents receive the message is 0.6 if e-mail is used, 0.8 if letter is used and 1 if cellphone is used. What is the probability that the parents receive the message?

A head of school contacts parents by e-mail, letter or cellphone with probabilities 0.4, 0.1 and 0.5 respectively. The probability parents receive the message is 0.6 for e-mail, 0.8 for letter and 1 for cellphone, so the overall probability of receiving the message is 0.82. Given that the parents received the message, what is the probability that it was sent by e-mail?

A student travels to school by bus, car, or on foot with probabilities $\frac{1}{6}$, $\frac{1}{3}$, and $\frac{1}{2}$ respectively. The probability of being late is $\frac{1}{5}$ if by bus, $\frac{1}{4}$ if by car, and $\frac{1}{20}$ if on foot. What is the probability that the student is early (not late) for school?

A student travels to school by bus, car, or on foot with probabilities $\frac{1}{6}$, $\frac{1}{3}$, and $\frac{1}{2}$ respectively. The probability of being late is $\frac{1}{5}$ if by bus, $\frac{1}{4}$ if by car, and $\frac{1}{20}$ if on foot. Given that the student arrives early on a particular day, what is the probability, as a fraction in lowest terms, that the student travelled on foot?

An ice cream vendor's sales are 40% chocolate, 35% vanilla and 25% peach. The probability of a cone sale is 0.80 for chocolate, 0.60 for vanilla and 0.40 for peach. What is the overall probability that a randomly selected sale is by cone?

Whenever there is a power-cut, a school is equally likely (probability 1/3 each) to switch on one of its 3 generators A, B or C. The independent probabilities of a breakdown are 0.2 for A, 0.3 for B and 0.25 for C. For a randomly chosen day with a power-cut, what is the probability that there was a generator breakdown?

Using the same power-cut scenario (each generator equally likely to be switched on with probability 1/3; independent breakdown probabilities 0.2 for A, 0.3 for B, 0.25 for C), given that there was a generator breakdown, what is the probability that it was generator C?

An unbiased die with faces marked 1, 2, 2, 3, 3, 3 is rolled twice. If X is the total score on the two rolls, what is P(X = 4), as a fraction out of 36?

A manufacturing plant uses three machines, A, B and C, in its production process. The total daily output contributions of machines A, B and C are 40%, 45% and 15% respectively. It is known that 4% of the tins produced by A are defective, 3% of those produced by B are defective, and 1% of those produced by C are defective. What is the probability that one tin chosen at random from the day's production is defective?

A manufacturing plant uses three machines, A, B and C, contributing 40%, 45% and 15% of daily output respectively, with defect rates of 4%, 3% and 1% for A, B and C respectively. Given that a randomly chosen tin is defective, what is the probability, to 3 decimal places, that it came from machine B?

A school's lower sixth intake is 55% from its own O-level pupils and 45% from other schools. Of those who did O-level elsewhere, 90% pass A-level; of those who did O-level at the school, 70% pass. A recent A-level graduate is selected at random. What is the probability that this pupil passed A-level?

Using the same school (55% own O-level pupils, 45% from elsewhere; pass rates 70% and 90% respectively, overall pass probability 0.79), given that a graduate passed A-level, what is the probability that they did O-level outside the school?

An experiment has only two possible outcomes. The first outcome occurs with probability $p$ and the second outcome occurs with probability $p^2$. Find the value of $p$, correct to 3 decimal places.

A survey in a city shows that the probability a person is in favour of capital punishment is 0.55 and the probability a person is against it is 0.45. If two people are selected at random, find the probability that at least one of them favours capital punishment.

A roulette wheel contains 38 numbers of which 18 are red, 18 are black and 2 are green. When the wheel is spun it is equally likely to land on any of the 38 numbers, and successive spins are independent. The wheel is spun twice. Find the probability that the ball lands on red both times, giving your answer correct to 4 decimal places.

A bag contains 24 counters of which 6 are red, 8 are green and 10 are yellow. Three counters are taken from the bag at random without replacement.