S1 – Probability Practice Questions set 2
an 2002 Q4
The employees of a company are classified as management, administration or production.
The following table shows the number employed in each category and whether or not they
live close to the company or some distance away.
Live close
Management
Administration
Production
6
25
45
Live some
distance away
14
10
25
An employee is chosen at random.
Find the probability that this employee
(a) is an administrator,
(2 marks)
(b) lives close to the company, given that the employee is a manager. (2 marks)
Of the managers, 90% are married, as are 60% of the administrators and 80% of the
production employees.
(c) Construct a tree diagram containing all the probabilities.
(3 marks)
(d) Find the probability that an employee chosen at random is married. (3 marks)
An employee is selected at random and found to be married.
(e) Find the probability that this employee is in production.
(3 marks)
Nov 2002 Q2 and Q3
2.
There are 125 sixth-form students in a college, of whom 60 are studying only arts subjects, 40
only science subjects and the rest a mixture of both.
Three students are selected at random, without replacement.
Find the probability that
(a) all three students are studying only arts subjects,
(4)
(b) exactly one of the three students is studying only science subjects.
(3)
S1 – Probability Practice Questions set 2
3.
The events A and B are independent such that P(A) = 0.25 and P(B) = 0.30.
Find
(a) P(A  B),
(2)
(b) P(A  B),
(2)
(c) P(AB).
(4)
Jan 2004 Q4
4.
The events A and B are such that P(A) =
2
1
4
, P(B) =
and P(AB ) = .
5
2
5
(a) Find
(i) P(A  B),
(ii) P(A  B),
(iii) P(A  B),
(iv) P(AB ).
(7)
(b) State, with a reason, whether or not A and B are
(i) mutually exclusive,
(2)
(ii) independent.
(2)
S1 – Probability Practice Questions set 2
Jan 2004 Q6
6.
One of the objectives of a computer game is to collect keys. There are three stages to the
2
1
game. The probability of collecting a key at the first stage is , at the second stage is , and
3
2
1
at the third stage is .
4
(a) Draw a tree diagram to represent the 3 stages of the game.
(4)
(b) Find the probability of collecting all 3 keys.
(2)
(c) Find the probability of collecting exactly one key in a game.
(5)
(d) Calculate the probability that keys are not collected on at least 2 successive stages in a
game.
(5)
S1 – Probability Practice Questions set 2
Question
Number
Jan 2002
1.
(a)
Scheme
Marks
a) P(Admin) = 35/125 = 7/25 or 0.28
b) P(Close given Manager) = P(C ∩ M) ÷ P(M) = 6/20 =3/10 or 0.3
c) First branches, P(Manager) = 20/125, P( Admin) 35/125,
P(Production) = 70/125;
Second branches: manager and married 18/20, not 2/20; Admin
and married 21/35, not 14/35; Production and married 56/70, not
14/70.
d) P(Married) = 20/125 x 18/20 + 35/125 x 21/35 + 70/125 x 56/70 =
95/125 or 0.76.
e) P(P given M) = P(P ∩ M) ÷ P(M) = 56/95 or 0.589
(b)
Nov 2002
2.
(a)
60A, 40S, 2M
P(all only arts) =
B1
60 59 58 3422



 0.10769.....
125 124 123 31775
(b) P(exactly one only science) = 3 
=
40 85 84


125 124 123
2856
= 0.44940…..
6355
M1 A1 A1 (4)
B1
M1 A1
(3)
(7 marks)
3.
(a) P(A  B) = P(A)P(B) = 0.25  0.30 = 0.075
M1 A1
(b) P(A  B) = P(A) + P(B) – P(A  B) = 0.25 + 0.30 – 0.075
M1
= 0.475
(c) P(AB) =
PA  B P(A)  PA  B
=
1  PB
P(B)
=
0.25  0.075
1  0.3
= 0.25
A1
(2)
(2)
M1
M1 A1ft
A1
(4)
(8 marks)
S1 – Probability Practice Questions set 2
Question
Mark Scheme
Jan 2004
4. (a) (i)
P(A  B ' )  P(A/B' ) P(B' ) 
(ii)
Marks
4 1
4
2
Use of
 

5 2 10 5 P (A/B' )P (B' )
P (A  B)  P(A)  P(A  B' )

(iv)
(b) (i)
(ii)
P (A  B)  P(A)  P(B) - P(A  B)

2 1
 0
5 2

9
10
P (A/B)  P
M1
2 2

5 5
=0
(iii)
M1
A1
(A  B)
0
P(B)
A1
M1
A1
B1
(7)
since P (A  B)  0 seen
A and B are mutually exclusive
B1
B1
(2)
Since P (A/B)  P (A) or equivalent
A and B are NOT independent
B1
B1
(2)
S1 – Probability Practice Questions set 2
Question
Mark Scheme
Marks
6. (a)
¼
½
K
¾
¼
⅔
⅓
K
K
K
¾
½
K
K
K
K
K
¼
K
½
½
K
¾
K
¼
K
¾
Tree
with
correct
number
of
branches
2 1
,
3 3
1 1 1 1
, , ,
2 2 2 2
1 3 3
, ...
4 4 4
M1
A1
A1
A1
(4)
K
1

0.08 3 ; M1 A1
12 ;
0.0833
(b)
2 1 1
2
1
 

P (All 3 Keys) = 3 2 4 = 24 12
(c)
 2 1 3 1 1 3 1 1 1
P (exactly 1 key)               
 3 2 4 3 2 4 3 2 4
added
10 5

= 24 12
3
(2)
triples M1
Each correct
A1
10
5
A1
24 ;
12 ; A1
A1
(5)

0.416 ; 0.417
(d)
P (Keys not collected on at least 2 successive
stages)
 2 1 3 1 1 1 1 1 3
          
 3 2 4  3 2 4 + 3 2 4
10 5


24 12
3
triples M1
added
A1
Each correct
A1
10 5
24 ; 12 ;

0.416 ; 0.417
A1
A1
(5)