  Probability

This is the measure of the likelihood of occurrence or a required outcome happening. It is the “chance” of an event happening. For example, a student might ask himself while preparing for an example, “what is the probability that I will score a hundred percent?” This means that the student is asking himself, what chance he has of scoring 100 out of 100.

Probability is often represented as a fraction = number of required outcomes

number  of possible outcomes

For the example of the student mentioned earlier, the boy has just 1required outcome which is to score 100% whilst the possible outcomes are 2 i.e. he either scores a 100% (1) or not (1)

So the probability of scoring 100% = ½

1. When an event is certain to happen, then the probability is 1 e.g. if Ade is 5years old this year, the probability that he would be 6 years old next year is 1.

2. When an event is certain not to happen, the probability is zero e.g. if Ade is 5years old now, the probability that he would be 8 years old next year is 0.

So if p is the probability that an event would happen and q is the probability that an event would not happen, then p + q = 1 è q = 1 – p. Thus if p is the probability of occurrence of a desired outcome, then the probability that the desired outcome would not occur is 1 – p.

Example:

A dice has six faces numbered 1 to 6. If the dice is rolled once, find the probability of

(a) obtaining the number  6 (b) obtaining the number 10 (c) not obtaining the number 6

(d) obtaining one of the numbers 1, 2, 3, 4, 5, 6

Solution:

(a)   Probability of obtaining 6 = Number of  6s on face of die =     1

Total numbers of figures on die   6

(b)   Probability of obtaining 10 is 0 since there is no 10 on the face of the die

(c)   Probability of not obtaining 6 = 1 – probability of obtaining 6 = 1 – 1/6 = 5/6

(d)   Since the numbers on the face of the die are 1, 2, 3, 4, 5 and 6, we can ONLY have one of these numbers so the probability is 1.

Factorisation

Factorisation means writing an expression in terms of its factors.  When an expression with two or more parts is being factorised, we remove all the factors common to the parts and place these factors outside the bracket leaving only the remainders in the brackets.

Example: Factorise the following

i.              3abx +  5adx = ax (3b + 5d)     ;  Reason: a & d are common

ii.            3d2e – 8d2 = d2 (3e – 8);   : d2  is common

iii.           -18fg – 12g = -6g(3f + 2);   : -6g is common

Indices

An index (plural – indices) is a short form of writing powers of a number. Example 53 = 5x5x5

The use of indices is guided by some simple laws which are given below.

1. Multiplication law

ax x ay =ax + y

Example: 41 x 43 = 41 + 3 = 44

1. Division law:  ax ÷ay =ax – y

Example: 43 ÷ 43 = 41 + 3 = 44

1. Power of Zero: a0 = 1 for any a ≠ 0

1000 = 1, 5.350 = 1

1. Negative power: a-x = 1/a

Example: 3-2 = 1/32 = 1/9

EXAMPLES: Simplify the following

a.    104 x 105 = 104 + 5 = 109

b.    M8÷ M5 = M8-5 = M3

c.    A-2 ÷B0 =  1/A2 ÷ 1 = 1/A2

Tests and Exercises

1. Simplify 102 x 105  (a) 10-3 (b) 103 (c) 107 (d) 1010

Guideline: 102 x 105 = 102+5 = 107

1. A card is picked at random from a pack of 52 playing cards. What is the probability that is a 6? (a) 1/52 (b) 1/13 (c) 3/26 (d) ¼

Guideline: In a pack of playing cards, there are 4 of every number, therefore

Probability of having 6 =            Number of 6      ____   = 4 =   1

Total number of playing cards   52    13

1. 10 + 15b = x (2 + 3b). What is x? (a) 2 (b)  3 (c) 5 (d) none of above

Guideline:  Since x is outside the bracket, it must be a common factor of 10 & 15b i.e. 5

1. A student is picked at random from a class containing 17 boys and 13 girls. What is the probability that the student is a girl? (a) 1/30 (b) 1/13 (c)13/30 (c) 13/17

Guideline:

Probability of picking a girl = Number of girls     =      13                = 13

Number of students          (13 + 17)          30

1. Which of the following is NOT a factor of 4x2y? (a) xy2 (b) 2x2y (c) x2y (d) 4x2y

Guideline:

Option A has y2 which cannot be obtained from 4x2y so A is the answer