Discrete random variables

The number of text messages, X, that a phone receives in a randomly chosen minute has the probability distribution below, where a is a constant.

A restaurant takes table bookings for groups of 1, 2, 3 or 6 people. The size of a randomly chosen booking is modelled by the random variable X with probability function P(X = x) = k/x for x = 1, 2, 3, 6, where k is a constant.

The number of strokes, X, that a club golfer takes on a short hole has the cumulative distribution function F(x) shown below.

A computer program picks a whole number X at random from 1, 2, 3, …, 8, each number being equally likely.

The number of goals, X, scored by a hockey team in a randomly chosen match is modelled by the probability distribution shown in Figure 1. Each probability lies on a gridline.

A commuter travels by train on 4 days each week. The number of days, X, in a randomly chosen week on which the train is late has the cumulative distribution function F(x) shown in Figure 1. The value of F at each step lies on a gridline.

Figure 1 shows a spinner at a school fair. When it is spun, the probability that it stops in a sector is proportional to the angle of that sector at the centre. The score, X, is the number shown in the sector where it stops.

The six cards in Figure 1 are shuffled and placed face down. One card is chosen at random and X is the number on it. The random variable U is defined by X = 5U.

A bag contains 4 red counters and 2 green counters. Two counters are taken from the bag at random, one after the other, without replacement. The random variable X is the number of green counters taken.

The number of attempts, X, that a climber needs to complete a route is modelled by P(X = x) = k(5 − x)² for x = 1, 2, 3, 4, where k is a constant.

In a puzzle app, the level X that a new player reaches on the first day takes the values 1, 2, 3, 4. The cumulative distribution function of X is F(x) = kx(x + 2) for x = 1, 2, 3, 4, where k is a constant.

The daily profit, X, in hundreds of pounds, of a market stall has the probability distribution below, where a and b are constants.

The number of passengers, X, on a minibus trip has E(X) = 9.2 and Var(X) = 4.36. Each passenger pays a fare of £3.50 and each trip costs £12 to run, so the profit on a trip, in pounds, is R = 3.5X − 12.

The 40 tickets in a raffle are numbered 21, 22, 23, …, 60. One ticket is drawn at random, and X is the number on it.

A goalkeeper faces two penalty kicks in a practice session. The tree diagram in Figure 1 shows the probabilities of the goalkeeper saving each kick. The random variable X is the number of kicks saved.

The number of people, X, in a car arriving at a car park is modelled by the probability distribution shown in Figure 1, where k is a constant. The car park charges £4 per car plus £1.50 per person, so the charge for a car, in pounds, is C = 1.5X + 4.

Figure 1 shows a circular target at a fair, made from three concentric circles of radii 10 cm, 20 cm and 30 cm. A dart thrown by a beginner always lands on the target and is equally likely to land at any point of it, so the probability of landing in a region is proportional to its area. The score, X, is 10, 5 or 2 as shown.

Spinner A has three equal sectors numbered 1, 2, 3 and spinner B has four equal sectors numbered 1, 2, 3, 4. Both are fair and are spun once, independently. The random variable X is the positive difference between the two scores (0 if they are equal). Figure 1 shows the value of X for each of the 12 equally likely outcomes.

The number of faults, X, found in a weekly inspection of a machine takes the values 0, 1, 2 and 3. Figure 1 shows the cumulative distribution function F(x) of X, where a and b are constants. The graph is not drawn to scale: the levels a and b are labelled on the vertical axis, and their values are to be found from the information below, not read from the graph. It is known that E(X) = 1.9 and that P(X = 3) = 2P(X = 1).

A game uses two six-sided dice, each numbered 1 to 6. The score X on the fair die and the score Y on a biased die have the distributions shown in Figure 1. The probabilities for the biased die lie on gridlines.

In a game, a fair four-sided die numbered 1, 2, 3, 4 is rolled until a 4 is obtained, with a maximum of three rolls: the player stops after the third roll whatever it shows. The random variable X is the number of rolls made.

The number of loaves, X, bought by a customer at a bakery is modelled by P(X = x) = a + bx for x = 1, 2, 3, 4, where a and b are constants. The mean number of loaves bought is 2.75.

A fair spinner has n equal sectors numbered 1, 2, …, n. The score X on one spin has a discrete uniform distribution. It is known that E(X²) = 63.

A student answers three quiz questions, A, B and C. The probabilities of answering them correctly are 0.9, 0.7 and 0.5 respectively, independently of each other. The random variable X is the number of questions answered correctly.

Figure 1 shows the six balls in a tub at a school fair. A player takes two balls at random from the tub, without replacement. The random variable X is the sum of the numbers on the two balls.

Figure 1 shows a game at a fair. A ball is released at the top of the board and meets one pin in each of the four rows. At each pin it is equally likely to go to the left or to the right, independently of what happened at earlier pins, and it then falls into one of the slots A to E. The number of points scored, X, is shown on each slot.

The number of parcels, X, delivered to an office in a randomly chosen hour takes the values 0, 1, 2, 3 and 4. Figure 1 shows part of the probability distribution of X: the probabilities shown lie on gridlines, and P(X = 1) = p and P(X = 3) = q are not drawn. It is known that E(X²) = 5.65.

In a computer game, a player opening a chest receives X coins. Figure 1 shows the cumulative distribution function F(x) of X. Every step of the graph lies on a gridline.