On this page you can find examination questions from the topic of statistics and probability
During week 1, a group of 60 athletes were asked to record the amount of water, X litres, that they consumed in that week. Here are the results
\(\sum x=1470\)
\(\sum x^2=36\ 132.6\)
Calculate
a. the mean of X
b. the standard deviation of X
During week 2, as part of a programme to improve their performance, the athletes were instructed to drink 10% more water. Assuming that they do this, find for week 2
c. the new mean
d. the new standard deviation
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The graph below shows the scores in the SL Mathematics Analysis and Approaches for at Goodenough High School. There are 20 students, the minimum score was 2 and the mean score is 4.45
How many grade 4s and how many grade 7s were there?
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The cumulative frequency graph gives information about the lengths, in minutes, of 80 telephone calls.
a. Find the median length of a phone call
b. Find the interquartile range of the length of a phone call
c. Find the number of phone calls that were more than 10 minutes in length
d. The frequency table below shows the lengths of the 80 phone calls. Find values a, b and c.
e. These data contain some outliers. How many outliers are there?
f. Calculate an estimate of the mean length of a phone call
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The world record times in seconds for the women’s 100m sprint from 1970 onwards are given below
Use your calculator to write down
a) \(\bar{x}\) , the mean year
b) \(\bar{y}\) , the mean time
c) \(r\), Pearson’s product-moment correlation coefficient
The equation of the regression line y on x is y = ax + b
d) Find the values of a and b for these data
e) Show that \((\bar{x},\bar{y})\) lies on this line
f) Use the regression line to estimate the world record time in 2024
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The table below shows the test scores of SL students following a six-week revision period using studyib.
a) Work out r, Pearson’s correlation coefficient
The y on x regression line is y = ax + b
b) Find a and b
c) A student scores 80 marks before revision. Use the regression line to estimate the score after revision
The x on y regression line is x = cx + d
d) Find c and d
e) A student scores 90 marks after revision. Use the regression line to estimate the score before revision
f) Find the point of intersection of the y on x and the x on y regression lines
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The following table shows the hand spans and the heights of eight basketball players on a team
The relationship between x and y can be modelled by the x on y line of regression x = ay + b
- Find the values of a and b
- Write down the correlation coefficient
- Another basketball player is 193cm tall. Use this regression line to estimate the handspan of this player.
Another player is 180cm tall. Use this regression line to estimate the handspan of this player.
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The following table shows the age in days (x) and length of new-born babies in cm (y).
The relationship between the variables is modelled by the regression line with equation y = ax + b
a) Find the values of a and b
b) Write down the correlation coefficient
c) Use your equation to estimate the length of a baby that is 40 days old
d) Use your equation to estimate the length of a baby that is 150 days old
e) Use your equation to estimate the age of a child that is 54 cm long.
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Two events A and B are such that P(A) = 0.35 and P(B) = 0.6 and \(P(A\cup B)\) = 0.74
a) Find \(P(A\cap B)\)
b) Determine whether A and B are independent
c) Find \(P(A'\cap B)\)
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Katniss is practising archery. She fires three arrows at a target. The probability that she hits the target with her first arrow is 0.7. Whenever she hits the target, her confidence increases so that the probability that she hits the target on her next attempt increases by 0.1. Whenever she misses the target, the probability reduces by 0.1.
a) Complete the probability tree for Katniss’s three attempts.
b) Calculate the probability that she hits the target with two attempts.
c) Find the probability the she hits the target on at least one attempt.
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Events A and B are such that P(A) = 0.4 and \(P(A\cup B)\) = 0.7
Find P(B) if A and B are independent
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Alphonse and Bettina are playing a game. A bag contains 2 yellow beads and 3 red beads. They take it in turns to pick a bead from the bag at random. Alphonse goes first. If Alphonse picks a yellow bead, he wins and the game stops. If he picks a red bead, he replaces the bead and it is Bettina’s turn. If Bettina picks a red bead, she wins and the game stops. If she picks a yellow bead, she replaces the bead and it is Alphonse’s turn again.
a) Find the probability that Alphonse wins on his first turn.
b) Show that the probability that Alphonse wins on his second turn is \(\frac{12}{125}\)
c) The game continues until one of the player wins. What is the probability that Alphonse wins the game?
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A box contains 25 tickets. x tickets are gold, the rest are silver. Two tickets are selected at random.
a) Show that the probability of selecting two gold tickets is \(\frac{x^2-x}{600}\)
b) Find the probability of selecting two tickets of the same colour.
c) The probability of selecting two tickets of the same colour is twice the probability of selecting two tickets of a different colour. Find how many gold tickets there are.
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Henri Tarr travels to school by bike \(\frac{3}{5}\) of the week and by car the rest of the time.
If he travels by bike, the probability that he is late is \(\frac{1}{8}\).
If he travels by car, the probability that he is late is \(\frac{3}{8}\).
a) Copy and complete the following tree diagram
b) Find the probability that Henri goes to school by car and is late for school.
c) Find the probability that he is late for school.
d) Given that he is late, find the probability that he travels to school by car.
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A and B are independent events. P(A) = 0.3 and P(B) = 0.4
a) Find \(P(A'\cap B')\)
b) Hence find P(A'|B')
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In the Olympic Games, it is believed that 5% of sprinters are taking anabolic steroids. A test is developed that gives a positive result for an athlete who has taken anabolic steroids in the last month in 90% of cases. The test gives a positive result for an athlete who has not taken anabolic steroids in the last month in 15% of cases. A sprinter in chosen at random.
a) Find the probability that she/he tests positive anabolic steroids
b) Given that the athlete tests positive, find the probability that the athlete has taken anabolic steroids in the last month.
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The discrete random variable X has probability function
P(X = x) = k(16 – x²) for x = 0, 1, 2, 3
- Find the value of the constant k
- Find P(1 ≤ X < 3)
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The random variable X has probability function
a) Find the value of k
b) Work out \(P(X\ge 2)\)
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Two boxes each contain three cards.
The first box contains cards labelled 1,3 and 5
The second box contains cards labelled 2, 4 and 6.
In a game, a player draws one card at random from each box and his score, X, is the sum of the numbers on the two cards.
a) Complete the following probability distribution
b) Work out E(X)
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The following table shows the probability distribution of a discrete random variable X
If X represents the return from a game. Find a and b if the game is fair.
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On average, it is found that 5% of AirPods* made on a production line are faulty.
a) Find the probability that in a random sample of 10, there are
i) No faulty AirPods
ii) more than one faulty set of AirPods
b) A sample of n sets of AirPods is taken from the production line. If the probability that there is at least one faulty AirPod is more than 75%, find the smallest possible value of n
* Fictitious data. I’m sure that AirPods production is very reliable!
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Monster Energy drink cans are advertised to contain 500ml. During the production, the volume X in ml in cans of the drink can be modelled by a normal distribution with mean 504 and variance 10.
a) For a randomly selected can, work out P(X > 500).
b) Cans are put in packs of 6. Find the probability that at least 5 cans are have a volume of at least 500ml.
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A glass contains 5 green sweets and m sweets of other colours. A sweet is taken at random.
a) Write down the probability that the sweet is green.
b) The sweet is replaced in the glass and the process is repeated a further three times. Each time, it is noted whether a green sweet is taken. The variance of the number of green sweets taken over the whole process in calculated to be 0.75.
Work out how many sweets in total there are in the glass.
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The probability density function of X is given by
\(f\left(x\right)=\left\{\begin{matrix}ax^n\\0\\\end{matrix}\ \ \ \ \ \ \ \right.\begin{matrix}0\le x<1\\otherwise\\\end{matrix}\)
a) Show that a = n + 1
b) Given that E(X) = 0.75, find a and n
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The continuous random variable X has a probability density function given by
\(f\left(x\right)=\left\{\begin{matrix}a\\asin\left(\frac{\pi x}{4}\right)\\0\\\end{matrix}\ \ \ \ \ \ \ \right.\begin{matrix}0\le x<2\\2\le x<4\\otherwise\\\end{matrix}\)
a) Draw a sketch of f
b) Show that the value of \(a=\frac{\pi}{2\pi+4}\)
c) Find E(X)
d) Find the exact value of the median of X
e) Find \(P(X\le2|X\le3)\)
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The continuous random variable X has a probability density function given by
\(f\left(x\right)=\left\{\begin{matrix}k\bullet a r c o s\left(x\right)\\0\\\end{matrix}\ \ \ \ \ \ \ \right.\begin{matrix}-1\le x<1\\otherwise\\\end{matrix}\)
a) Draw a sketch of f
b) State the mode of X
c) Find \(\int a r c o s\left(x\right)dx\)
d) Find E(X)
e) Find Var(X)
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The random variable X is normally distributed with a mean of 100 and a standard deviation of 15.
a) Write down P(X < 120)
b) Write down P(95 < X < 105)
c) P(X < c) = 0.6. Write down the value of c
d) Find the interquartile range of X
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The weights of female adult penguins found on a remote island are normally distributed with a mean of 3.15 kg. It is found that 81% of these penguins weigh more than 3 kg. The weights of the penguins have a standard deviation of \(\sigma\).
a) Find the standardized value for 3kg.
b) Hence, find the value of \(\sigma\).
It is estimated that there are 4000 female adult penguins on the island. Penguins are considered underweight if they weigh less than 2.9 kg.
c) Find the estimated number of underweight female penguins on the island.
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It is known that 2% of Filips lightbulbs have a life of less than 1000 hours and 10% have a life less than 1100 hours. It can be assumed that lightbulb life is normally distributed with a mean of \(\mu\) and a standard deviation of \(\sigma\)
a) Find the value of \(\mu\) and the value of \(\sigma\).
b) Find the probability that a randomly selected Filips lightbulb will have a life of at least 1200 hours.
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The amount of caffeine, X in mg, found in a cup of tea can be modelled by the normal distribution \(X\sim N(26,5)\)
a) Find the probability that a randomly selected cup of tea contains at least 27 mg of caffeine.
b) 10 cups of tea are randomly selected. Find the probability that more than 5 of these cups has at least 24 mg of caffeine.
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How much of Statistics and Probability Examination Questions HL have you understood?
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