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Exam-Style Questions.Problems adapted from questions set for previous Mathematics exams. |
1. | IB Standard |
Mr Willis goes to school (he's a Maths teacher) five times each week during term time. On any given day, the probability that he goes on an orange bus is \( \frac13 \).
(a) Write down the expected number of times that Mr Willis goes to work at his school on an orange bus in one week.
In one week, find the probability that he goes to work on an orange bus
(b) on exactly three days;
(c) on at least two days.
2. | IB Standard |
A small digital storage manufacturer makes novelty USB drives which are given away at conferences. The probability that a drive is defective is 0.03. The factory tests a random sample of 200 drives.
(a) Find the mean number of defective drives in the sample.
(b) Find the probability that there are exactly ten defective drives in the sample.
(c) Find the probability that there is at least one defective drive in the sample.
3. | IB Analysis and Approaches |
The probability of winning a gold coin in a game of chance is 0.6. The game is played six times. Find the probability of winning:
(a) at least two gold coins;
(b) a fourth gold coin on the sixth game.
4. | IB Analysis and Approaches |
A factory manufactures switches. It is known that the probability that a switch is found to be defective is 0.04. A random sample of 50 switches is tested.
(a) Find the probability that there is at least one defective switch in the sample.
(b) Given that there is at least one defective switch in the sample, find the probability that there are at least three defective switches.
5. | IB Standard |
A famous pole vaulter finds that she is successful on 85% of her attempts at clearing five metres. Her Wikipedia entry confirms that her success rate is 85%.
A field sports day event consists of 12 five-metre vaults.
a) Find the probability that she is successful on all 12 attempts.
b) Find the probability that she is successful on at least 8 attempts.
She can earn £50 in prize money for every successful vault
c) What can she expect her winnings to be in a one day event?
The probability that another vaulter is successful at clearing five metres 3 times out of 12 is 0.166.
d) What are the possible value(s) of his success rate?
6. | IB Analysis and Approaches |
The travelling times by train, \( T \) minutes, between two towns can be modelled by a normal distribution with a mean of 55 minutes and a standard deviation of \( \sigma \) minutes.
(a) Given that 8% of the times are longer than 68 minutes, find the value of \( \sigma \).
(b) Find the probability that a randomly selected train journey will have a duration of more than 75 minutes.
(c) Given that a train journey between the two towns takes longer than an hour, find the probability that it takes less than 70 minutes.
On a particular day, there are 42 trains scheduled to travel between these two towns.
(d) Find the expected number of train journeys that will have a duration of more than an hour.
(e) Find the probability that more than 10 of the journeys on this particular day will have a duration of more than one hour.
7. | IB Applications and Interpretation |
Bostock and Chandler play one game of Ultimate Noughts and Crosses online each day they are in quarantine.
The probability that Bostock wins a game is three times the probability that Chandler wins a game. It is not possible to have a draw.
(a) Find the probability that on any given day Bostock will win the game.
The quarantine lasts for 30 days.
(b) Find the probability the Bostock will win 20 times.
(c) Find the probability Chandler will win at most 9 times.
8. | IB Standard |
At the south end of Playfair Amusement park are two games of chance, Beat The Goalie and Basketball Bounce.
For the Goalie game, the probability of winning is 0.1. If someone plays this game nine times:
(a) Find the probability that they win exactly three times.
For the Basketball game, the probability of winning is \(x\). If someone plays it nine times.
(b) Write down an ex
(c) Hence, find the values of \(x\) such that the probability of winning exactly three games is 0.2.
9. | IB Standard |
The length of Costlow's bâtard bread loaves in centimetres is normally distributed with mean \( \mu \). The following table shows probabilities for values of \(L\).
| Length (\(L\)) | \(L \lt 30\) | \(30 \le L \le 42\) | \(L \gt 42\) |
|---|---|---|---|
| P(X) | 0.07 | 0.86 | \(k\) |
(a) Calculate the value of \(k\).
(b) Show that \( \mu \) = 36.
(c) Find P(\(L \gt 39\)).
The loaves are displayed in baskets of twelve. Any loaves with a length less than 31cm are classified as short.
(d) Find the probability that a basket of loaves selected at random contains at most one loaf that is short.
(e) Each Costlow supermarket has 40 baskets of loaves. One of the Costlow supermarkets is selected at random. Find the expected number of baskets in this supermarket that contain at most one loaf that is too short.
(f) Find the probability that at least 28 baskets in this supermarket contain at most one loaf that is too short.
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