SUBJECT: MATHEMATICS

CLASS: SS 2

DATE:

TERM: 3rd TERM

REFERENCE BOOKS

New General Mathematics SSS2 by M.F. Macrae etal.
Essential Mathematics SSS2 by A.J.S. Oluwasanmi.
Exam Focus Mathematics.

WEEK NINE

TOPIC: PROBABILITY

CONTENTS

MUTUALLY EXCLUSIVE AND INDEPENDENT EVENTS

APPLICATION OF TREE DIAGRAM IN SOLVING PROBLEMS

Mutually Exclusive Event

Mutually exclusive events are events which cannot together at the same time. One event will pave way for the other, in such a case the separate probability are added together probabilities are added to give the combined probability.

Additional Law of Probability

If event A,B,C…. are mutually exclusive, the probability of A or B or C or…. Happening is the sum of their individual probabilities.

P(A) + P(B) + P(C) + ……..

Note: use the addition law to solve problems that contains the word or or either/or.

Worked Examples:

A bag contains 3 red balls, 4 blues balls 5 white balls and 6 black balls. A ball is picked at

Random, what is the probability that it is either:

Red or blue
Blue or black
Red, white or blue
Blue, white or black
Neither red nor

Solution

P(R) = 3/18 P(B) = 4/18 P(W) = 5/18, P(BK) =6/18

(a)Pro(either red or blue) = 3/18 + 4/18 = 7/18

(b) Pro(blue or black) = 4/18 6/18

= 10/18 = 5/9

(c)P(red white or blue) = P(r) +P(w) + P(blue)

= 3/18 + 4/18 + 5/18

=12/18 = 2/3

(d) P(blue, white or black) = P(blue) + P(white) + P(black)

= 4/18 + 5/18 + 6/18

= 15/18 = 5/6

(e)P(neither Red or blue) = P(R or blue)1

= 1-P(R or blue)

= 1- 7/18

= 11/18

Worked Example 2

A letter is choosen at random from the word “COMPUTER” what the probability that it is

(a) either in the word cut or in the word ROPE

(b) neither in the word MET nor in the word UP?

Solution:

n(s) = 8

(a) P (either word CUT or ROPE) =

P (CUT) + P(ROPE)

= 3/8 + 4/8

= 7/8

(b) P(MET +UP)1 = 1- 5/8 = 3/8

Evaluation

F={2, 3, 7} and T = {10, 20, 30, 40}

(a) If one element is selected at random, from F, write down the probability that it is odd.

(b) If one element is selected at random from T, write down the probability that it is a multiple of 5

INDEPENDENT EVENT

Independent event are event which have no effect on each other. In such cases the

Separate probabilities are multiplied to give the combined probability.

Product Law

If event A, B, C is independent, the probability of A and B and C and …. Happening is

the product of their individual probabilities P(A) x P(B) x P(c) ……

Note: use the product law to solve problems that contains the words “and” or both/and

Worked Example: A coin is tossed and a die is then thrown what is the probability of getting a

head and a perfect square

Solution

P(H and perfect square)

P(H) = ½

(Perfect square = (1,4)

n (perfect square) = 2

n(s) = 6

P (perfect square) = 2/6 = 1/3

∴ P (H and perfect square) = `1/2 x 1/3 =

=1/6

WorkedExample 2:

A bag contains 3 black balls and 2 white balls

(a) A ball is taken from the bag and then replaced, A second ball is chosen, what is the probability that

(i) They are both black

(ii) One is black and one is white

(iii) at least one is black

(iv) at most one is black

Solution.

With Replacement

i P(BB) = 3/5 x 3/5 = 9/25

ii Probabilities that one is black and one is white = P(BW) or P(WB)

P(one white one black) = P(BW) + P(WB)

= 3/5 x 2/5 + 2/5 x 3/5

P(BW) or P(WB) = 6/25 + 6/25 = 12/25

iii Prob ( at least one is black) = P( both are black) + P(one is black)

= P(BW) + P (WB) +P(BB)

= 12/25 + 9/25

= 21/25

iv At most one is black means either one is black or non is black i.e one is black or both are white.

P(at most one black) = P(BW) + P(WB) + P(WW)

= 6/25 + 6/5 + 4/25

= 16/25

Evaluation

A box contains 5 blue balls 3 green balls.

(a) A ball is taken from the box and then replaced.A second is chosen is chosen,what is the probability that (i)they are both blue (ii)one blue and one is green (iii)at least one is blue

GENERAL EVALUATION

1.A box contains ten marbles,seven of which are black and three are red.Three marbles are drawn one after the other without replacement.Find the probability of choosing a) one red,one black and one red marble(in that order).

b) two black marbles
c) at least two black marbles
d) at most two black marbles

READING ASSIGNMENT

NGM SSS2,page115-116, Exercise 11b, 1-10.

WEEKEND ASSIGNMENT

Objectives

Two fair dice tossed together at once find the probability that the sum of the outcome is at least 10(a) 1/12 (b) 3/15 (c) 5/36 (d) 2/5

2 form a box containing 2 Red, 6 white and 5 black balls, a ball is randomly selected , what istheprobability that the selected ball is black.(a) 5/12 (b) 5/13 (c) 4/5 (d) 7/13

A bag contains 3 red, 4 black and 5 green identical balls, 2 balls are picked at random one after the other without replacement, find the prob that one is red and the other is green

(a) 5/22 (b) 7/23 (c) 15/132 (d) 12/13

A bag contains 3 white, 6 red and 5 blue identical balls, a ball is picked at random from the bag, what is the prob. that it is either white or blue? (a) 9/14 (b) 5/14 (c) 4/7 (d) 6/7

A bag contains red, black and green identical balls,a ball is picked andReplaced at 100 times.The table below shows the result of the 100 trails, What is the probability of picking a green ball.

Colour	Red	Black	Green
No. of occurrence	54	30	16
(a)21/25	(b)16	(c)4/25	(d)1/3

THEORY

A box contains 5 blue balls, 3 black balls and 2 red balls of the same size. A ball is selected at random, from the box and then replaced. A second ball is then selected, find the probability of obtaining.

(a) Two red balls

(b) Two blue balls or 2 black balls

Solve the same problem if it is without replacement.

Lesson Notes By Weeks and Term - Senior Secondary School 2