Download IBSL Prob Actual Exams since 2004 No Solution IBSL

IBSL Statistic and Probability
Actual Exams Since 2004
1. Dumisani is a student at IB World College.
(2004 p2)
7
The probability that he will be woken by his alarm clock is 8.
1
If he is woken by his alarm clock the probability he will be late for school is 4.
3
5
If he is not woken by his alarm clock the probability he will be late for school is .
Let W be the event “Dumisani is woken by his alarm clock.”
Let L be the event “Dumisani is late for school.”
a) Copy and complete the tree diagram below.
[4 marks]
b) Calculate the probability that Dumisani will be late for
school.
[3 marks]
c) Given that Dumisani is late for school what is the
probability that he was woken by his alarm clock?
[4 marks]
2. The table below shows the marks gained in a test by a group of students.
Mark
1
2
3
4
5
Number of
students
5
10
P
6
2
(2004 p1)
The median is 3 and the mode is 2. Find two possible values of p.
3
7
3. Let A and B be events such that 𝑃(𝐴) = 4 and 𝑃(𝐴 ∪ 𝐵) = 8.
a) Calculate 𝑃(𝐴 ∩ 𝐵).
b) Calculate 𝑃(𝐴|𝐵).
c) Are the events A and B independent? Give a reason for your answer.
(2004 p1)
4. The cumulative frequency curve below shows the marks obtained in an examination by a group
of 200 students.
(2004 p1)
a) Use the cumulative curve to complete the frequency table below.
Mark (x)
0 ≤ 𝑥 < 20 20 ≤ 𝑥 < 40 40 ≤ 𝑥 < 60 60 ≤ 𝑥 < 80
Number of
22
students
80 ≤ 𝑥 < 100
20
b) Forty percent of the students fail. Find the pass mark.
5. The table below shows the subjects studied by 210 students at a college.
History
Science
Art
Totals
Year 1
50
15
45
110
Year 2
35
30
35
100
(2005 p2)
Totals
85
45
80
210
a) A student from the college is selected at random.
[6 marks]
Let A be the event the student studies Art.
Let B be the even the student is in Year 2.
i)
ii)
iii)
Find 𝑃(𝐴).
Find the probability that the student is a Year 2 art student.
Are the events A and B independent? Justify your answer.
b) Given that a history student is selected at random, calculate the probability that the student is in
Year 1.
[2 marks]
c) Two students are selected at random from the college. Calculate the probability that one
student is in Year 1, and the other is in Year 2.
[4 marks]
6) The 45 students in a class each recorded the number of whole minutes, x, spent doing
experiments on Monday. The results are ∑ 𝑥 = 2230.
(2005 p1)
a) Find the mean number of minutes the students spent doing experiments on Monday.
Two new students joined the class and reported that they spent 37 minutes and 30 minutes
respectively.
b) Calculate the new mean including these two students.
6. Two unbiased 6-sided dice are rolled, a red one and a black one. Let E and F be the events
E: the same number appears on both dice;
F: the sum of the numbers is 10.
Find
a) 𝑃(𝐸);
b) 𝑃(𝐹);
c) 𝑃(𝐸 ∪ 𝐹).
7. Consider the four numbers a, b, c, d with 𝑎 ≤ 𝑏 ≤ 𝑐 ≤ 𝑑, where 𝑎, 𝑏, 𝑐, 𝑑 ∈ ℤ.
(2006 p1)
The mean of the four numbers is 4. The mode is 3. The median is 3. The range is 6.
Find the value of a, of b, of c, and of d.
8. The heights of a group of students are mutually distributed with a mean of 160 cm and a
standard deviation of 20 cm.
(2006 p1)
a) A student is chosen at random. Find the probability that the student’s height is greater than
180cm.
b) In this group of students, 11.9% have heights less than d cm. Find the value of d.
9. Let A and B be independent events such that 𝑃(𝐴) = 0.3 and 𝑃(𝐵) = 0.8.
a) Find 𝑃(𝐴 ∩ 𝐵).
b) Find 𝑃(𝐴 ∪ 𝐵).
c) Are A and B mutually exclusive? Justify your answer.
(2006 p1)
10. The heights of boys at a particular school follow a normal distribution with a standard deviation
of 5 cm. The probability of a boy being shorter than 153 cm is 0.705.
(2007 p1)
a) Calculate the mean height of the boys.
b) Find the probability of a boy being taller than 156 cm.
11. The cumulative frequency graph below shows the heights of 120 girls in a school. (2007 p1)
a) Using the graph
i) Write down the median;
ii) Find the interquartile range.
b) Given that 60% of the girls are taller than a cm, find the value of a.
2
5
1
4
7
8
12. Consider the events A and B, where 𝑃(𝐴) = , 𝑃(𝐵′ ) = and 𝑃(𝐴 ∪ 𝐵) = . (2007 p1)
a) Write down P(B).
b) Find 𝑃(𝐴 ∩ 𝐵).
c) Find 𝑃(𝐴|𝐵).
13. A four-sided die has three blue faces and one red face. The die is rolled.
(2008 p1)
Let B be the event a blue face lands down, and R be the even a red face lands down.
a) Write down
[2 marks]
i)
P(B).
ii)
P(R).
b) If the blue face lands down, the die is not rolled again. If the red face lands down, the die is
rolled over again. This is represented by the following tree diagram, where p, s, t are
probabilities.
[2 marks]
Find the value of p, of s, and of t.
c) Guiseppi plays a game where he rolls the die. If a blue face lands down, he scores 2 and is
finished. If the red face lands down, he scores 1 and rolls one more time. Let X be the total
score obtained.
[3 marks]
3
i)
Show that 𝑃(𝑋 = 3) = 16.
ii)
Find 𝑃(𝑋 = 2).
di)
Construct a probability distribution table for X.
dii)
Calculate the expected value for X.
e)
If the total score is 3, Guiseppi wins $10. If the total score is 2, Guiseppi gets nothing.
Guiseppi plays the game twice. Find the probability that he wins $10.
[5 marks]
[4 marks]
14. A box contains a large number of biscuits. The weights of biscuits are normally distributed with
mean 7 g and standard deviation 0.5 g.
(2008 p2)
a) One biscuit is chosen at random from the box. Find the probability that this biscuit
i)
Weighs less than 8 g;
ii)
Weighs between 6 g and 8 g.
[4 marks]
b) Five percent of the biscuits in the box weigh less than d grams.
[5 marks]
i)
Copy and compete the following normal distribution, to represent this information,
by indicating d, and shading the appropriate region.
ii) Find the value of d.
c) The weights of biscuits in another box are normally distributed with mean 𝜇 and standard
deviation 0.5 g. It is known that 20% of the biscuits in this second box weigh less than 5 g.
Find the value of 𝜇.
[4 marks]
15. Factory makes switches. The probability that a switch is defective is 0.04. The factory tests a
random sample of 100 switches.
(2008 p2)
a) Find the mean number of defective switches in the sample.
[2 marks]
b) Find the probability that there are exactly six defective switched in the sample. [2 marks]
c) Find the probability that there is at least one defective switch in the sample.
[3 marks]
16. In a school with 125 girls, each student is tested to see how many sit-up exercises (sit-ups) she
can do in one minute. The results are given in the table below.
(2008 p2)
Number of
sit-ups
15
16
17
18
19
20
Number of
students
11
21
33
q
18
8
Cumulative number
of students
11
32
p
99
117
125
ai) Write down the value of p.
aii) Find the value of q.
[3 marks]
b.
Find the median number of sit-ups.
[2 marks]
c)
Find the mean number of sit-ups.
[2 marks]
17. The letters of the word PROBABILITY are written on 11 cards as shown below. (2009 p1)
Two cards are drawn at random without replacement.
Let A be the event the first card drawn is the letter A.
Let B be the even the second card drawn is the letter B.
a) Find 𝑃(𝐴).
b) Find 𝑃(𝐵|𝐴).
c) Find 𝑃(𝐴 ∩ 𝐵).
[1 mark]
[2 marks]
[3 marks]
18. A fisherman catches 200 fish to sell. He measures the lengths, 𝑙 cm of these fish, and the results
are shown in the frequency table below.
(2009 p2)
Length 𝑙 cm
Frequency
0 ≤ 𝑙 < 10
0 ≤ 𝑙 < 10
0 ≤ 𝑙 < 10
0 ≤ 𝑙 < 10
0 ≤ 𝑙 < 10
0 ≤ 𝑙 < 10
0 ≤ 𝑙 < 10
30
40
50
30
33
11
6
a) Calculate an estimate for the standard deviation of the lengths of fish.
[3 marks]
b) A cumulative frequency diagram is given below for the lengths of the fish.
Use the graph to answer the following.
i)
ii)
Estimate the interquartile range.
Given that 40% of the fish have a length more than k cm, find the value of k.
[6 marks]
In order to sell the fish, the fisherman classifies them as small, medium or large.
Small fish have a length less than 20 cm.
Medium fish have a length greater than or equal to 20 cm but less than 60 cm.
Large fish have a length greater than 60 cm.
c) Write down the probability that a fish is small.
[2 marks]
The cost of a small fish is $4, a medium fish is $10, and a large fish is $12.
d) Copy and complete the following table, which give a probability distribution for the cost $X.
[2 marks]
Cost $X
𝑃(𝑋 = 𝑥)
e) Find 𝐸(𝑋).
4
10
0.565
12
[2 marks]
19. In any given season, a soccer team plays 65% of their games at home.
(2009 p2)
When the team plays at home, they win 83% of their games.
When they play away from home, they win 26% of their games.
The team plays one game.
[4 marks each]
a) Find the probability that the team wins the game.
b) If the team does not win the game, find the probability that the game was played at home.
20. A random variable X is distributed normally with mean 450 and standard deviation 20. (2009 p2)
a) Find 𝑃(𝑋 ≤ 475).
b) Given that 𝑃(𝑋 > 𝑎) = 0.27,
find 𝑎.
[2 marks]
[4 marks]
21. Consider the events A and B, where 𝑃(𝐴) = 0.5, 𝑃(𝐵) = 0.7 and 𝑃(𝐴 ∩ 𝐵) = 0.3. (2010 p1)
The Venn diagram below shows the events A and B, and the probabilities p, q and r.
a) Write down the value of
i) p;
ii) q;
iii. r.
b) Find the value of 𝑃(𝐴|𝐵′ ).
c) Hence, or otherwise, show that the events A and B are not independent.
[3 marks]
[2 marks]
[1 mark]
22. The weights of players in a sports league are normally distributed with a mean of 76 kg.
It is known that 80% of the players have weights between 68 kg and 82 kg.
The probability that a player weighs less than 68 kg is 0.05.
(2010 p2)
a) Find the probability that a player weighs more than 82 kg.
b) i. Write down the standardized value, z, for 68 kg.
ii. Hence, find the standard deviation of weights.
[2 marks]
[4 marks]
To take part in a tournament, a player’s weight must be within 1.5 standard deviations of
the mean.
c) i. Find the set of all possible weights of players that take part in the tournament.
ii. A player is selected at random. Find the probability that a player takes part in the
tournament.
[5 marks]
Of the players in the league, 25% are women. Of the women, 70% take part in the
tournament.
d) Given that a player selected at random takes part in the tournament, find the probability
that the selected player is a woman.
[4 marks]
23. Evan likes to play two games of chance, A and B.
(2010 p2)
[7 marks]
For game A, the probability that Evan wins is 0.9. He plays game A seven times.
a) Find the probability that he wins exactly four games.
[2 marks]
For game B, the probability that Evan wins is p. He plays game B seven times.
[2 marks]
b) Write down an expression, in terms of p, for the probability that he wins exactly four games.
c) Hence, find the values of p such that the probability that he wins exactly four games in 0.15.
24. In a group of 16 students, 12 take art and 8 take music. One student takes neither art nor music.
The Venn diagram below shows the events art and music. The values p, q, r and s represent
numbers of students.
(2011 p1)
a) i. Write down the value of s.
ii. Find the value of q.
iii. Write down the value of p and of r.
[5 marks]
b) i. A student is selected at random. Given that the student takes music, write down the
probability the student takes art.
ii. Hence, show that taking music and taking art are not independent events. [5 marks]
c) Two students are selected at random, one after the other. Find the probability that the first
student takes only music and the second takes only art.
[4 marks]
25. The following frequency distribution of marks has mean 4.5.
Mark
1
2
3
4
5
6
7
Frequency 2
4
6
9
x
9
4
a) Find the value of x.
b) Write down the standard deviation.
(2010 p2)
[4 marks]
[2 marks]
26. The probability distribution of a discrete random variable X is given by
𝑥2
𝑃(𝑋 = 𝑥) = 14 , 𝑥 ∈ {1,2, 𝑘}, 𝑤ℎ𝑒𝑟𝑒 𝑘 > 0.
a) Write down𝑃(𝑋 = 2).
b) Show that 𝑘 = 3.
c) Find 𝐸(𝑋).
(2011 p1)
[1 mark]
[4 marks]
[2 marks]
27. A company uses two machines, A and B, to make boxes. Machine A makes 60% of the boxes.
80% of the boxes made by machine A pass inspection.
(2011 p2)
90% of the boxes made by machine B pass inspection.
A box is selected at random.
a) Find the probability that it passes inspection.
[3 marks]
b) The company would like the probability that a box passes inspection to be 0.87. Find the
percentage of boxes that should be made by machine B to achieve this.
[4 marks]
28. A random variable X is distributed normally with a mean of 20 and variance 9.
(2011 p2)
a) Find 𝑃(𝑋 ≤ 24.5).
[3 marks]
b) Let 𝑃(𝑋 ≤ 𝑘) = 0.85.
i)
Represent this information on the following diagram.
ii)
Find the value of k.
29. Events A and B are such that 𝑃(𝐴) = 0.3, 𝑃(𝐵) = 0.6, 𝑎𝑛𝑑 𝑃(𝐴 ∪ 𝐵) = 0.7.
[5 marks]
(2012 p1)
The values of q, r, s and t represent probabilities.
a)
b)
c)
Write down the value of t.
i. Show that 𝑟 = 0.2.
ii. Write down the values of q and s.
i. Write down 𝑃(𝐵′ ).
ii. Find 𝑃(𝐴|𝐵′ ).
30. A box holds 240 eggs. The probability that an egg is brown is 0.05.
a) Find the expected number of brown eggs in the box.
b) Find the probability that there are 15 brown eggs in the box.
c) Find the probability that there are at least 10 brown eggs in the box.
[1 mark]
[3 marks]
[3 marks]
(2011 p2)
[2 marks]
[2 marks]
[3 marks]
31. The probability of obtaining “tailes” when a biased coin is tossed is 0.57. The coin is tossed ten
times. Find the probability of obtaining
(2012 p2)
a) at least four tailes;
b) the fourth tail on the tenth toss.
[4 marks]
[3 marks]
32. A random variable X is normally distributed with 𝜇 = 150 𝑎𝑛𝑑 𝜎 = 10.
Find the intergauartile range of X.
(2013 p2)
[7 marks]
33. The ages of people attending a music concert are given in the table below.
Age
Frequency
Cumulative
Frequency
15 ≤ 𝑥 < 19 19 ≤ 𝑥 < 23
14
26
14
40
23 ≤ 𝑥 < 27
52
92
27 ≤ 𝑥 < 31
52
P
a) Find the value of p.
(2012 p1)
31 ≤ 𝑥 < 35
16
160
[2 marks]
The cumulative frequency diagram is given below.
b)
Use the diagram to estimate
i. the 80th percentile;
ii. the interquartile range.
[5 marks]
34. The histogram below shows the time T seconds taken by 93 children to solve a puzzle. (2012 p2)
The following is a frequency distribution for T.
Time
Frequency
45 ≤ 𝑇 < 55
7
55 ≤ 𝑇 < 65
14
65 ≤ 𝑇 < 75
p
a) i. Write down the value of p and of q.
ii. Write down the median class.
75 ≤ 𝑇 < 85
20
85 ≤ 𝑇 < 95
18
95 ≤ 𝑇 < 105
q
105 ≤ 𝑇 < 115
6
[3 marks]
b) A child is selected at random. Find the probability that the child takes less than 95 seconds
to solve the puzzle.
[2 marks]
Consider the class interval 45 ≤ 𝑇 < 55.
c) i. Write down the interval width.
ii. Write down the mid-interval value.
[2 marks]
d) Hence, for an estimate for the
i. mean;
ii. standard deviation.
[4 marks]
Jon assumes that T is normally distributed and uses this to estimate the probability that a child takes less
than 95 seconds to solve the puzzle.
c) Find John’s estimate.
2 marks]
4
5
35. Bill and Andrea play two games of tennis. The probability that Bill wins the first game is .
5
If Bill wins the first game, the probability that e wins the second game is 6.
2
If Bill loses the first game, the probability that he wins the second game is 3.
(2014 p1)
a) Copy and complete the following tree diagram.
[3 marks]
b) Find the probability that Bill wins the first game and Andrea wins the second game. [2]
c) Find the probability that Bill wins at least one game.
[4]
d) Given that Bill wins at least one game, find the probability that he wins both games. [5]
36. Jar A contains three red marbles and five green marbles. Two marbles are drawn from the jar,
one after the other, without replacement.
(2013 p1)
a) Find the probability that
i. none of the marbles are green;
ii. exactly one marble is green.
[5 marks]
b) Find the expected number of green marbles drawn from the jar.
[3 marks]
Jar B contains six red marbles and two green marbles. A fair six-sided die is tossed. If the score is 1 or 2,
a marble is drawn from jar A. Otherwise, a marble is drawn from jar B.
c) i. Write down the probability that the marble is drawn from jar B.
[2 marks]
ii. Given that the marble is drawn from jar B, write down the probability that it is red.
d) Given that the marble is red, find the probability that it was drawn from jar A.
[6 marks]
37. The weights in grams of 80 rats are shown in the following cumulative frequency diagram.
(2014 p2)
a) i. write down the median weight of the rats.
[4]
ii. Find the percentage of rats that weigh 70 grams or less.
The same data is presented in the following table.
Weights w grams
Frequency
0 ≤ 𝑤 ≤ 30
p
30 < 𝑤 ≤ 60
45
b) i. Write down the value of p.
ii. Find the value of q.
60 < 𝑤 ≤ 90
q
90 < 𝑤 ≤ 120
5
[4 marks]
c) Use the values from the table to estimate the mean and standard deviation of the weights. [3]
Assue that the weights of these rats are normally distributed with the mean and standard deviation
estimatd in part (c).
d) Find the percentage of rats that weight 70 grams or less.
[2 marks]
e) A sample of five rats is chosen at random. Find the probability that at most three rats weight 70
grams or less.
[3 marks]
38. Consider the following cumulative frequency table.
X
5
15
25
35
45
Frequency Cumulative
frequency
2
2
10
12
14
26
p
35
6
41
a)
Find the value of p.
b) Find
i. the mean;
ii. the variance.
(2013 p2)
[2]
[4]
39. Celeste wishes to hire a taxicab from a company which has a large number of taxicabs. The
taxicabs are randomly assigned by the company.
(2014 p1)
The probability that a taxicab is yellow is 0.4.
The probability that a taxicab is a Fiat is 0.3.
The probability that a taxicab is yellow or a Fiat is 0.6.
Find the probability that the taxicab hired by Celeste is not a yellow Fiat.
[6 marks]
40. A company makes containers of yogurt. The volume of yogurt in the containers is normally
distributed with a mean of 260 ml and standard deviation of 6 ml.
(2015 p2)
A container which contains less than 250 ml of yogurt is underfilled.
a) A container is chosen at random. Find the probability that it is underfilled.
[2]
The company decides that the probability of a container being underfilled should be reduced to
0.02. It decreases the standard deviation to 𝜎 and leaves the mean unchanged.
b) Find 𝜎.
[4]
The company changes to the new standard deviation, 𝜎, and leaves the mean unchanged. A
container is chosen at random for inspection. It passes inspection if its volume of yoguart is
between 250 and 271 ml.
c) i. Find the probability that it passes inspection.
[6]
ii. Given that the container is not underfilled, find the probability that it passes inspection.
d) A sample of 50 containers is chosen at random. Find the probaility that 48 or more of the
containers pass inspection.
[4]
41. A discrete random variable X has the following probability distribution.
X
𝑃(𝑋 = 𝑥)
0
3
10
1
4
10
2
2
10
(2015 p1)
3
p
a) Find p.
b) Find 𝐸(𝑋).
42. Ann and Bob play a game where they each have an eight-sided die. Ann’s die has three green
faces and five red faces; Bob’s die has four green faces and four red faces. They take turns
rolling their down die and note what colour faces up. The first player to roll green wins. Ann rolls
first. Part of a tree diagram of the game is shown below.
(2015 p1)
a) Find the probability that Ann wins on her first roll.
[2]
5
4
3
b) i. The probability that Ann wins on her third roll is 8 𝑥 8 𝑥 𝑝 𝑥 𝑞 𝑥 8.
Write down the value of p and of q.
3
ii. The probability that Ann wins on her tenth roll is 8 𝑟 𝑘 𝑤ℎ𝑒𝑟𝑒 𝑟 ∈ ℚ, 𝑘 ∈ ℤ.
Find the value of r and of k.
c) Find the probability that Ann wins the game.
[6]
[7]
43. The following table shows the average number of hours per day spent watching television by
seven mothers and each mother’s youngest child.
(2015 p2)
Hours per day that a mother
wathces televison (x)
Hours per day that her child
wathces televison (y)
2.5
3.0
3.2
3.3
4.0
4.5
5.8
1.8
2.2
2.6
2.5
3.0
3.2
3.5
The relationship canbe modelled by the regression line with equation 𝑦 = 𝑎𝑥 + 𝑏.
a) i. Find the correlation coefficient.
ii. Write down the value of a and of b.
[4]
Elizabeth wathces television for anaverage of 3.7 hours per day.
b) Use your regression line to predict the average number of hours of television watched per
day by Elizabeth’s youngest child. Give your answer correct to one decimal place. [3]