Practise Probability & Statistics 1 through data representation, counting, probability, discrete random variables and normal-distribution models using exact exam methods.
The random variable X takes the value x with probability , where k is a constant and x takes the values -2,1,2,3 only.
Draw up the probability distribution table for X, giving the probabilities as numerical fractions.
Find E(X).
Find .
A coin is biased so that the probability of obtaining a head when it is thrown is 0.4. The coin is thrown repeatedly until the first head is obtained.
Find the probability that the first head is obtained on the 5th throw.
Find the probability that the first head is obtained after the 6th throw.
The Splash Club has 26 members, of whom 16 are swimmers and 10 are divers. No member is both a swimmer and a diver. The club committee consists of 6 of these 26 members.
In how many ways can the club committee be selected if it must include at least 2 swimmers and at least 2 divers?
Chen has three boxes. Box A contains 5 counters, of which 3 are white and 2 are yellow. Box W contains 4 red marbles and 3 blue marbles. Box Y contains 5 red marbles and 3 blue marbles.
Chen chooses one counter at random from box A. If the counter is white, he chooses two marbles from box W, at random and without replacement. If the counter is yellow, he chooses two marbles from box Y, at random and without replacement.
Draw a fully labelled tree diagram to illustrate this information, including all the probabilities.
Find the probability that one red marble and one blue marble are obtained.
Find the probability that a white counter is chosen from box A, given that one red marble and one blue marble are obtained.
In a group of 25 athletes, there are 8 sprinters, 5 hurdlers and 12 throwers.
Find the number of different ways in which a team of 6 athletes can be selected if it consists of at least 3 sprinters, at most 2 hurdlers and at most 1 thrower.
A group of 8 athletes chosen from the group of 25 athletes consists of 1 sprinter, 3 hurdlers and 4 throwers. These 8 athletes stand in a row.
How many different arrangements of the 8 athletes are there if the 3 hurdlers do not stand all together?
How many different arrangements of the 8 athletes are there in which there are at least two athletes between any two hurdlers?
Last Saturday, a cycling competition for teams of 11 cyclists took place. For each cyclist, the time taken to complete the course was recorded to the nearest minute. The times taken by the cyclists from two teams, the Linnets and the Puffins, are shown in the following table.
Draw a back-to-back stem-and-leaf diagram to represent this information, with Linnets on the left-hand side.
Find the interquartile range of the times taken by the Linnets.
On the grid below, draw a box-and-whisker plot to represent the information for the Linnets and the Puffins.
Make one comparison between the times taken by the Linnets and the times taken by the Puffins.
The mass of grapes sold per day by a large shop can be modelled by a normal distribution with mean 28 kg . On 10 % of days less than 16 kg of grapes are sold.
Find the standard deviation of the mass of grapes sold per day.
The mass of grapes sold on any day is independent of the mass sold on any other day.
12 days are chosen at random.
Find the probability that less than 16 kg of grapes are sold on more than 2 of these 12 days.
In a random sample of 365 days, on how many days would you expect the mass of grapes sold to be within 1.3 standard deviations of the mean?
The heights, in cm, of the 11 basketball players in each of two clubs, the Amazons and the Giants, are shown below.
State an advantage of using a stem-and-leaf diagram compared to a box-and-whisker plot to illustrate this information.
Represent the data by drawing a back-to-back stem-and-leaf diagram with Amazons on the left-hand side of the diagram.
Find the interquartile range of the heights of the players in the Amazons.
Four new players join the Amazons. The mean height of the 15 players in the Amazons is now 191.2 cm . The heights of three of the new players are and 190 cm .
Find the height of the fourth new player.
Find the number of different arrangements of the 10 letters in the word ZOOLOGICAL in which the three Os are together and the two Ls are not next to each other.
Find the number of different arrangements of the 10 letters in the word ZOOLOGICAL in which there are exactly 5 letters between the two Ls.
Two letters are chosen at random from the 10 letters in the word ZOOLOGICAL. Find the probability that these two letters are different.
The times taken by Obi to walk to work each morning are normally distributed with mean 14.8 minutes and standard deviation 1.5 minutes.
Obi walks to work 5 days a week for 45 weeks in a year.
Find the probability that, on a randomly chosen day, Obi takes more than 15.6 minutes to walk to work.
On of days, Obi takes more than t minutes to walk to work.
Find the value of t.
On how many days in a year would you expect Obi to take within one minute of the mean time to walk to work?
