Mathematics 2025 · Oct/Nov

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 . [2] bTTTTuTTTTTTTib

Kayla has a bag containing 3 red marbles, 1 blue marble and 2 green marbles. She selects one marble from the bag at random and does not replace it in the bag. She repeats this process until she obtains a green marble. The random variable X is the number of marbles that she needs to select until she obtains a green marble.

Draw up the probability distribution table for X. [4] bTTTTuTTTTTTTsb

The heights of the 124 Senior members of the Giraf fes basketball club are normally distributed with mean 187.4 cm and standard deviation 6.4 cm.

How many members of the club would you expect to have heights within 5 cm of the mean? [4] bTTTTuTTTTTTT b The heights of the Junior members of the Giraf fes club are normally distributed with mean 172.7 cm and standard deviation v cm. 23% of these members have height less than 170.3 cm.

Find the value of v. [3] bTTTTuTTTTTTTdb

Gio has a pack of 18 cards. Ivy has a pack of x cards. Each card has a picture of a bus or a car or a train. The number of cards with each picture in the two packs is shown in the table. Bus Car Train Gio's pack 6 10 2 Ivy's pack x12- 9 3 One card is chosen at random from each pack. The probability that the two cards have pictures of buses on them is equal to twice the probability that the two cards have pictures of cars on them.

Write down an equation in terms of x and hence find the value of x. [4] bTTTTuTTTTTTTob

Find the probability that the two cards have pictures of the same type of vehicle on them. [3] bTTTTuTTTTTTTcb

The times of 240 competitors taking part in an event are recorded correct to the nearest minute. The results are summarised in the table. Time (minutes)1 - 1011 - 2021 - 2526 - 3031 - 50 Frequency 12 38 68 76 46

Draw a histogram to represent this information. [4] * 0000800000008 * bTTTTuTTTTTTTub

Calculate an estimate of the mean time taken by the 240 competitors. [3] bTTTTuTTTTTTTmb

For a randomly chosen person, their next birthday is equally likely to occur on any day of the week, independently of any other person's birthday.

Find the probability that, out of 10 randomly chosen people, none of them will have their next birthday on a Saturday or Sunday.

Find the probability that, out of 10 randomly chosen people, fewer than 3 will have their next birthday on a Wednesday. [3] bTTTTuTTTTTThTb

Use a suitable approximation to find the probability that, out of 392 randomly chosen people, more than 65 will have their next birthday on a Friday. [5] bTTTTuTTTTTThhb

Find the number of dif ferent 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 dif ferent arrangements of the 10 letters in the word ZOOLOGICAL in which there are exactly 5 letters between the two Ls. [3] bTTTTuTTTTTThib Two letters are chosen at random from the 10 letters in the word ZOOLOGICAL.

Find the probability that these two letters are dif ferent. [3] bTTTTuTTTTTThsb Additional page If you use the following page to complete the answer to any question, the question number must be clearly shown. bTTTTuTTTTTTh b BLANK PAGE * 0000800000015 * bTTTTuTTTTTThdb Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable ef fort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge Assessment International Education Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cambridgeinternational.org after the live examination series. Local Examinations Syndicate (UCLES), which is a department of the University of Cambridge. BLANK PAGE * 0000800000016 * bTTTTuTTTTTThob