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Exam-Style Questions.Problems adapted from questions set for previous Mathematics exams. |
1. | IB Analysis and Approaches |
The time in minutes taken to drive through a city at different times of the day can be modelled by a normal distribution with mean of 45 and standard deviation of 12. A commuter will be late for work if it takes more than an hour to drive through the city.
(a) Find the probability that the commuter is late for work.
(b) Estimate the number of days on which the commuter was late from random sample of 100 days.
(c) Given that the commuter is not late, find the probability that the cross-city journey took less than 25 minutes.
2. | 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.
3. | IB Standard |
The weights of players in a sports league are normally distributed with a mean of 75.2 kg, (correct to three significant figures). It is known that 75% of the players have weights between 67 kg and 80 kg. The probability that a player weighs less than 67 kg is 0.05.
(a) Find the probability that a player weighs more than 80 kg.
(b) Write down the standardized value, z, for 67 kg.
(c) Hence, find the standard deviation of weights.
To take part in a tournament, a player's weight must be within 1.5 standard deviations of the mean.
(d) Find the set of all possible weights of players that take part in the tournament.
(e) A player is selected at random. Find the probability that the player takes part in the tournament.
Of the players in the league, 22% are women. Of the women, 60% take part in the tournament.
(f) Given that a player selected at random takes part in the tournament, find the probability that the selected player is a woman.
4. | IB Analysis and Approaches |
The length, \(X\) minutes , of a certain category of online video is normally distributed with a mean of 28.
The probability that \(X\) is less than 20 is 0.213.
(a) Find \(P(20 \lt X \lt 28)\).
(b) Find the standard deviation of \(X\).
(c) Hence, find the probability that a video selected at random from this category lasts longer than 33 minutes
A random sample of 12 videos from this category are downloaded.
(d) How many of these videos could be expected to last longer than 33 minutes?
(e) Find the probability that exactly two of these videos last longer than 33 minutes.
(f) A video selected at random from the complete online collection of videos in this category has a running time of less than 20 minutes. Find the probability that its length is between ten and fifteen minutes.
5. | IB Analysis and Approaches |
The random variable X follows a normal distribution with mean \(\mu\) and standard deviation \(\sigma\).
(a) Find \( P(\mu - 1.25\sigma \lt X \lt \mu + 1.25\sigma) \).
The pineapples grown on a farm in Thailand have weights, in grams, that are normally distributed with mean \(\mu\) and standard deviation \(\sigma\). Pineapples are categorised as tiny, regular, super or giant, according to their weight. The following table shows the probability a pineapple grown on the farm is classified tiny, regular, super or giant.
| Size | Tiny | Regular | Super | Giant |
|---|---|---|---|---|
| Probability | 0.08 | 0.72 | 0.16 | 0.04 |
The maximum weight of a tiny pineapple is 895 grams.
The minimum weight of a giant pineapple is 1804 grams.
(b) Find the values of \(\mu\) and \(\sigma\).
One season a wholesaler purchased all the regular, super and giant pineapples from the farm.
Find the probability that a pineapple chosen at random from this purchase is categorized as
(c) regular.
(d) super.
(e) giant.
The wholesaler sells the pineapples at the following prices:
| Size | Regular | Super | Giant |
|---|---|---|---|
| Price | 40 ฿ | 60 ฿ | 80 ฿ |
The wholesaler pays the farm 8000 ฿ for the pineapples and assumes it will then sell them in exactly the same proportion as purchased from the farm.
(f) Find the minimum number of pineapples that must be sold so that the net profit for the supermarket is at least 4000 ฿.
6. | 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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