IBSL Unit Exam Review
Probability and Statistic
Name:
Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported
by working and/or explanation. In particular, solutions found from a graphic display calculator should be
supported by suitable working, e.g., if graphs are used to find a solution, you should sketch these as part
of your answer. Where an answer is incorrect, some marks may be given for a correct method, provided
this is shown by written working. You are therefore advised to show all working. Working may be
continued below the lines, if necessary.
1.
P
The letters of the word PROBABILITY are written on 11 cards as show below. (adapted
from IB 2009)
R
O
B
A
B
I
L
I
T
Y
Two cards are drawn at random without replacement.
Let A be the even the first card drawn is the letter A.
Let B be the event the second card drawn is the letter B.
a) Find P(A).
b) Find P( B A).
c) Find P ( A  B ).
2.
Bag A contains 2 red balls and 3 green balls. Two balls are chosen at random from the
bag without replacement. Let X denote the number of red balls chosen. The following
table shows the probability distribution for X.
(adapted from IB 2005)
X
P(X = x)
0
3
10
1
6
10
a) Calculate E (X), the mean number of red balls chosen.
2
1
10
[3 marks]
Bag B contains 4 red balls and 2 green balls. Two balls are chosen at random from bag B.
bi) Draw a tree diagram to represent the above information, including the probability of each
event.
[3 marks]
bii) Hence find the probability distribution for Y, where Y is the number of red balls chosen.
[5 marks]
A standard die with six faces is rolled. If a 1 or 6 is obtained, two balls are chosen from bag A,
otherwise two balls are chosen from bag B.
c) Calculate the probability that two red balls are chosen.
[5 marks]
d) Given that two red balls are obtained, find the conditional probability that a 1 or 6 was rolled
on the die.
[3 marks]
3.
In a game a player rolls a biased four-faced die. The probability of each possible score is
shown below.
(adapted from IB 2005)
Score
Probability
1
1
6
2
2
7
3
3
8
4
x
a) Find the value of x.
[2 marks]
b) Find E (X).
[3 marks]
c) The die is rolled twice. Find the probability of obtaining two scores of 3.
[2 marks]
4.
The probability of obtaining heads on a biased coin is
times. Find the probability of getting:
1
. Simba tosses the coin three
5
(adapted from IB 2005)
a) Three heads.
[2 marks]
b) Two heads and one tail.
[3 marks]
Lizzie plays a game in which she tosses the coin 15 times.
c) Find the expected number of heads.
[2 marks]
d) Lizzie wins $ 10 for each head obtained, and loses $ 6 for each tail. Find her expected
winnings.
[3 marks]
5.
A factory makes switches. The probability that a switch is defective is 0.04. The factory
tests a random sample of 100 switches.
(adapted from 2007 IB)
a) Find the mean number of defective switches in the sample.
[2 marks]
b) Find the probability that there are exactly six defective switches in the sample. [2 marks]
c) Find the probability that there is at least one defective switch in the sample.
6. A and B are events such that P(A) = 0.4 and P(B) = 0.3.
a) If A and B are mutually exclusive events, find P( A  B).
marks]
[3 marks]
(adapted from 1995 IB)
[2
b) If A and B are independent events, find P( A  B).
[2 marks]
c) If A and B are independent events, find P( A B).
[2 marks]
.
7.
Consider the expansion of the expression ( x 3  3x) 6 .
(adapted from 2007 IB)
a) Write down the number of terms in this expansion.
[1 mark]
b) Find the term in x12.
[3 marks]
8.
A six-sided die has four blue faces and 2 red faces. The die is rolled. Let B the event
of a blue face lands down, and R be the event a red face lands down. Write down:
(adapted from 2008 IB)
a) P(B).
[1 mark]
b) P(R).
[1 mark]
c) If the blue face lands down, the die is not rolled again. If the red face lands down, the die is
rolled once again. This is represented by the following tree diagram, where p, s, t are
probabilities.
B
p
B
t
R
s
R
Find the value of p, of s, and of t.
[2 marks]
Mojojojo plays a game where he rolls the die. If a blue face lands down, he scores 4 and is
finished. If the red face lands down, he scores 2 and rolls one more time. Let X be the total score
obtained.
d) Find P (X = 4).
[1 mark]
e) Find P (X = 6).
[2 marks]
f) Construct a probability distribution table for X.
[3 marks]
g) Calculate the expected value for X.
[2 marks]
h) If the total score is 6, Mojojojo wins $ 20. If the total score is 4 Mojojojo gets nothing.
Mojojojo plays the game twice. Find the probability that he wins exactly $ 20.
[4 marks]
9. Consider the events A and B, where P(A) = 0.5, P(B) = 0.7, P(A∩B) = 0.3.
The Venn diagram below shows the events A and B, and the probabilities p, q, and r.
(adapted from 2010 IB p1)
(a) Write down the value of
(i)
p;
(ii)
q;
(iii)
r.
[3 marks]
(b) Find the value of P(A│B’).
[2 marks]
(c) Hence, or otherwise, show that the events A and B are not independent. [1 mark]
10. The following frequency distribution of marks has mean 4.5. (adapted from 2010 IB p2)
1
Mark
Frequency 2
2
4
3
6
4
9
5
x
6
9
7
4
(a) Find the value of x.
[4 marks]
(b) Write down the standard deviation.
[2 marks]
11. Evan likes to play two games of chance, A and B.
(adapted from 2010 IB p2)
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.
(b) Write down an expression, in terms of p, for the probability that he wins exactly four
games.
[2 marks]
(c) Hence, find the values of p such that the probability that he wins exactly four games
is 0.15.
[3 marks]
12. 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.
(adapted from 2010 IB p2)
(a) Find the probability that a player weighs more than 82 kg.
[2 marks]
(b) (i) Write down the standardized value, z, for 68 kg.
(ii) Hence, find the standard deviation of weights.
[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 the 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]
13. The probability distribution of a discrete random variable X is given by
𝑥2
𝑃(𝑋 = 𝑥) =
, 𝑥 ∈ {1, 2, 𝑘}, where 𝑘 > 0.
14
(adapted from 2011 IB p1)
(a) Write down P(X = 2).
[1 mark]
(b) Show that k = 3.
[4 marks]
(c) Find E(X).
[2 marks]
14. 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.
(adapted from 2011 IB 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 that the student takes art.
(ii) Hence, show that taking music and taking art are not independent events.
[4 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 student takes only art. [4 marks]
15. A random variable X is distributed normally with a mean of 20 and variance 9.
(adapted from 2011 IB 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.
[5 marks]
16. A box holds 240 eggs. The probability that an egg is brown is 0.05.
(adapted from 2011 IB p2)
(a) Find the expected number of brown eggs in the box.
[2 marks]
(b) Find the probability that there are 15 brown eggs in the box.
[2 marks]
(c) Find the probability that there are at least 10 brown eggs in the box.
[3 marks]
17. A company uses two machines, A and B, to make boxes. Machine A makes 60% of the
boxes.
(adapted from 2011 IB p2)
80% of the boxes made by machine A pass inspection.
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]
18. The ages of people attending a music concert are given in the table below.
(adapted from IB 2012 p1)
15 ≤ x < 19 19 ≤ x < 23 23 ≤ x < 27 27 ≤ x < 31 31 ≤ x < 35
Age
14
26
52
52
16
Frequency
40
92
P
160
Cumulative 14
Frequency
(a) Find P.
The cumulative frequency diagram is given below.
[2 marks]
(b) Use the diagram to estimate
(i)
The 80th percentile;
(ii)
The interquartile range.
[5 marks]
19. Events A and B are such that P(A) = 0.3, P(B) = 0.6, and P( A  B). = 0.7
(adapted from 2012 IB p1)
The values q, r, s and t represent probabilities.
(a) Write down the value of t.
[1 mark]
(b) (i) Show that r = 0.2.
(ii) Write down the value of q and of s.
[3 marks]
(c) (i) Write down P(B’).
(ii) Find P(A│B’).
[3 marks]
20. The probability of obtaining “tails” when a biased coin is tossed is 0.57. The coin is
tossed ten times. Find the probability of obtaining
(adapted from 2012 IB p2)
(a) At least four tails;
[4 marks]
(b) The fourth tail on the tenth toss.
[3 marks]
21. The histogram below shows the time T seconds taken by 93 children to solve a puzzle.
(adapted from 2012 IB p2)
The following is the frequency distribution for T.
45 ≤ T < 55
55 ≤ T < 65
65 ≤ T < 75
75 ≤ T < 85
Time
14
P
20
Frequency 7
85 ≤ T < 95
95 ≤ T < 105
105 ≤ T < 115
18
Q
6
(a) (i) Write down the value of P and Q.
(ii) Write down the median class.
[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 find an estimate for the
(i)
mean;
(ii)
standard deviation.
[4 marks]
John 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.
(e) Find John’s estimate.
[2 marks]
22. Jar A contains three red marbles and five green marbles. Two marbles are drawn from the
jar, one after the other, without replacement.
(adapted from 2013 IB 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
(ii) Given that the marble was drawn from jar B, write down the probability that it is
red.
[2 marks]
(d) Given that the marble is red, find the probability that it was drawn from jar A.
[6 marks]
23. Consider the following cumulative frequency table.
X
Frequency Cumulative frequency
5
2
2
15
10
12
25
14
26
35
p
35
45
6
41
(a) Find the value of p.
(adapted from 2013 IB p2)
[2 marks]
(b) Find
(i)
The mean;
(ii)
The variance.
[4 marks]
24. A random variable X is normally distributed with µ = 150 and α = 10. Find the
interquartile range of X.
[7 marks]
(adapted from 2013 IB p2)