Lesson Notes By Weeks and Term - Senior Secondary School 2

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 EIGHT

TOPIC: PROBABILITY (EVENT AND OUTCOME)

CONTENT

1. Definition of terms 
2. Events and outcome (measuring probability)

    (a)    Experimental probability 

    (b)    Theoretical probability 

DEFINITION OF TERMS

(i)Event: When an experiment is performed two or more results or outcomes will be expected to happen. Each attempt is called a trial and the outcome of a trial and the outcome of a trial is called an event, usually denoted by E.

(ii)Random Experiment: A random experiment is a repetitive process which may result in any one of the possible outcomes of the experiment OR:

(iii)Sample space: The sample space of a random experiment is the set containing all the possible outcomes of the experiment OR:

Sample space is all the possible outcomes of a trail in an experiment usually denoted by S.   

(iv)The number of the points in a sample space n(s), and in an event, E is n(E).

Examples

1. When a coin is tossed twice, all the possible outcomes i.e. the sample space 

    S  =  {HH, HT,TH, TT}

    ∴ n(s) = 4

2. If a die is cast once, there are six outcomes.

    ∴ the sample space , S = {1, 2, 3, 4, 5, 6}

    ∴ n(S) = 6

    Suppose an event E that an even number is thrown,

    then E = {2, 4, 6} and n(E) = 3. 

3. A box contain 16 red, 6 white, and 18 blues balls.

    The sample spaces, S = {16 + 6 + 18) balls

    n (S) = 40

4. When a die is tossed twice, the outcome of the first toss S1 = (1, 2, 3, 4, 5, 6) does not influence the outcome of the second throw. S2 = (1, 2, 3, 4, 5, 6). The two outcomes are independent of each other. For instance, the chance of throwing a5 in the first toss is 1/6 does not influence the chance of the throw of 2 in the second toss (i.e. 1/6); they are Independent Event. 

Equally likely events: Two or more events are said to be equally likely to happen if the chance of occurrence of each of the same.e.g. 

1.In the throw of a die, there are six equally likely outcomes, S = {1, 2, 3, 4, 5, 6} the change of each occurring is 1 out of 6 c.c. 1/6.

2.From a pack of 52 cards, the chance of picking any of the cards at random is 1/52.

PROBABILITY

The probability of an event is the chance of its occurrence, that is the likelihood of the event happening with respect to the sample space.   

Prob. Of E =          number of elements in E___

        number of total elements in S

∴P(E) = n(E)

      n(S)

NOTE: Probability of an event lies between 0 and 1 i.e. O<P(E) <1

then the prob. that it will not occur is 1 – P(E).

EVALUATION

1. In a class of 27 boys and 12 girls, what is the probability of picking a girl.
2. A no is chosen at random from 40 to 50, find the probability that it is a prime number.
3. If all 2-digits numbers 00, 01, 02, …….99 are equally likely to be chosen, find the probability that a number picked at random has 5 as its first digit.

EXPERIMENTAL AND THEORETICAL PROBABILITY 

EXPERIMENTAL PROBABILITY

Experimental Prob =  no of required outcome

            no of possible outcome

Example

A die is rolled 200 times, the outcome obtained are shown below.

Find the experimental probability of obtaining (a) 6    (b) 2

(a)    P(6)    =    n(6)    =      32    =       4   

              n(S)200            25             = 0.16

(b)    P(2)    =    n(2)        =    30    =      3          =   0.15

            n(S)        200            20

Since experimental probability uses numerical records of past events to predicts the future, its predictions are not absolutely accurate, however the probability of throwing a 2 on a fair 6-sided die is 1/6, since any one of the 6 faces is equally alike. This is an example of theoretical probability.  

THEORETICAL PROBABILITY

Theoretical probability is the assumed value assigned to the occurrence of an event based on the assumption that each of the elements in the outcome are equally likely to happen i.e. by considering the physical nature of the given situation.  

Examples

Tola throws a fair six-sided die, what is the probability that she throws (a) a 9  (b) a 4  

(c) a no greater than 2       (d) an even no      (e) either 1, 2, 3, 4, 5, or 6?

Solution

9. Since the faces of a six sided die are numbers 1, 2, …6, it is impossible to throw a 9.

    ∴ P (9) = 0

1. There is a chance out of 6 chances of throwing 4

    ∴ P(4)  =  1/6 

1. S    =   {1, 2, 3, 4, 5, 6}   ,     n (S) = 6

    no > 2  =  {3, 4, 5, 6}   ,    n(<2)  =  4

    P (no >2)  =   n(no>2)    =    4/6           =     2/3

        n(s)   

1. There are 3 possible even number  S = {1, 2, 3, 4, 5, 6},     n (S) = 6

    even no = {2, 4, 6}    n (even) = 3

    P(even) =     n (even)    =        3/6    =  ½  

        n (S)       

1. Either 1, 2, 3, 4, 5, 6

    S = {1, 2, 3, 4, 5, 6}        n (S) = 6n  (r) = 6.

P(r)  =  n(r)=      6/6      =     1

                n(S)   

Example 2

A bag contains 3 red, 5 green and 7 white balls, if a ball is selected from the bag, what is the probability that the ball is green?

Total no. of balls, n(S) = 3 + 5 + 7 = 15

Event E = green balls ∴n(E) = 5

∴P(E) = n(E)      =    5    =   1

      n(S)            15        3

EVALUATION

Use the figure below to answer the following:

(a)    If a number is picked at random from the figure. What is the probability that it is:-

    (i)    Odd          (ii)    Prime     (iii)    even       (iv)    less than 10

    (v)    Exactly divisible by 3       (vi)    a perfect square   (vii)    a perfect cube?

(b)    If a row or column is picked at random from the figure. What is the probability that the total of its no is(i)    34         (ii)    35

GENERAL EVALUATION

1    A bag contains black balls, 3 green balls and 4 red balls, A ball is picked form the bag at random, what is the probability that it is

    (a) Black     (d) yellow    (c) Green     (d) not black (d) either black ore red

2    A school contains 357 boys and 323 girls, if a student is chosen at random, what is the probability that a girl is chosen.

READING ASSIGNMENT 

NGM SSS2, page113-114, exercise11a, numbers 1-12. 

WEEKEND ASSIGNMENT

OBJECTIVE

1    What is the probability of throwing a number greater than 4 with a single fair die. 

              (a)  ½    (b)     1/3   (c)    5/6   (d) 2/3

2    A number is chosen at random from the set (11, 12, 13, ….25) what is the probability that    the number is odds?(a)  7/15    (b) 8/15       (c)    1/4    (d)   3/4

3    A box contains 8 blues 6 yellow and 10 green balls , one all is picked at random from the box, what is the probability that the ball is yellow. (a)   1/3 (b)½ (c) 3/4 (d)    5/12

4    A coin is tossed twice, what is the probability of obtaining at least a head  

(a) 3/4    (b) 1/3    (c) 2/5    (d)  1/2

5    A letter is chosen at random from the word PROBABILITY, what is the probability that the letter is a vowel? (a) 3/11(b)  4/11   (c) 5/11   (d)  6/11

THEORY

1    Two groups of male students X and Y cast their votes in an election of an officer; he results are as shown in the table below:

1. How many students participate in the election?
2. If a student in favour of the  officer is selected, what is the probability that he is from group X?
3. A student is choosen at random, what is the probability that he is against the officer?

2    A ltter is choose at random from the alphabet. Find the probability that it is (a) M (b) not A or Z (c) Either P, Q, R, or S (d) One of the letters of NIGERIA.