Mathematics Notes


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


            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?


            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   


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

(c)        If one element is selected at random, from FUT write down the probability of 42 or a multiple of 4


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


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 =                  


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


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     


      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



1.         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

3.         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

4.         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

5.         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.

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


1.         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

(c)        One black and one red ball in any order.

2.         Solve the same problem if it is without replacement.

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