Back to: Further Mathematics SS2

**INDEPENDENT AND DEPENDENT EVENTS**

Independent event: Tow or more events are said to be independent when the occurrence of one event does not affect the occurrence of the other events in any way. Hence, the events can occur independently. E .g obtaining a 6 in a single throw of a die and obtaining a tail in an event of coin. This leads to the **MULTIPLICATION RULE.**

*Multiplication rule:If E _{1, }E_{2, }E_{3, }….E_{n }*

**are independent events then Pr(E**

_{1}nE_{2}…nE_{n})=Pr(E1) x Pr(E2) x …Pr (En)When two events are combined with words such as; **AND, BOTH.** The probabilities of the events are multiplied.

An event is said to independent when picking is done with replacement.

** Dependent Event:** Two of more events are dependent when the occurrence of event 1 affects the occurrence of the other event (s). It is dependent when it is done without replacement.

** Conditional Probability: **This is the application of the multiplication rule.

**Example:**

1. A class consists of 8 men and 7 ladies. Two students were selected randomly to represent the class in a debate. Find the probability that two students selected are (a) both ladies (b) both men (c) of the same sex (d) of different sexes.

Solution:

Total students = 8+ 7 =15

n(men) = 8, n(ladies) = 7

a) Pr(both ladies) = 7/15 x 6/14 =1/5

b) Pr(both men) = 8/15 x 7/14 =4/15

c) Pr(same sex) = Pr(m & m) or Pr (L & L)

4/15 + 1/5 = 7/15

d) Pr (different sexes) = P (m & L) or P (L or m)

= (8/15 x7/14) + (7/15 x 8/14)

= 4/15 + 4/15 = 8/15

2. Two marbles are selected randomly, from a box containing 5 red, 7 white and 8 blue marbles one after the other without replacement. Find the Probability that: (a) both are red (b) one is white and the other is blue. (c0 both are of the same colour.

**Solution:**

Total Marbles = 5 + 7 + 8 = 20

N(R) = 5, n(w) =7, n(B) = 8

(a) Pr (both red) = P(RR) = 5/20 x 4/19 =1/19

(b) Pr (one white, one blue) = The arrangement is important.

Pr(wB) or P(Bw) = (7/20 x 8/19) + (8/20 x 7/19)

= 14/95 + 14/95 = 28/95

(c) Pr(same colour) = Pr(RR) or Pr(WW) or Pr (BB)

= (5/20 x 4/19) + (7/20 x 6/19) + (8/20 x 7/19)

= 20/380 + 42/380 + 56/380

= 118/380 =59/180

**EVALUATION :**

1. A box contains 2 white and 3 blue identical marbles. If two marbles are picked at random, one after the other, without replacement, what is the probability of picking two marbles of different colour?

2. A ball is picked at random from a bag containing 5 green balls, 3 white balls and 2 black balls. What is the probability that it is either green or black?

**FURTHER EXAMPLE:**

1. Two gubernational aspirants A and B in two different states of Nigeria have probabilities 2/9 and 4/11 respectively of winning in an impending election. Fid the probability that: (a) both of them win in their respective states; (b) both of them lose in their respective states; (c) at least one of them wins in his state.

**Solution:**

Aspirants: A B

Pr (A wins) + pr (A losing) = 1, Pr(Bwins) + Pr(B losing) = 1

**Pr (A **losing) = 1 – 2/9 Pr (B losing) = 1- 4/11

**=7/9 7/11 **

(a) **Pr**(both winning) =2/9 x 4/11

= **8 /99**

(b) Pr(both winning) = 7/9 x 7/11 = 49/99

(c) Pr (at least one wins) = Pr (A wins & B loses) or Pr(Bwins&Aloses) or Pr (A wins & B wins)

**= (2/9 x 7/11) + (4/11 x 7/9) + (2/9 x 4/11)**

**= 11/99 + 28/99 + 8/99 = 50/99.**

**Evaluation: **The probabilities that Bala and Uzoamaka will pass an examination are given as 0.8 and 0.9 respectively. Find the probability that: (a) both of them fail the examination (b0 at least one of them will pass this examination.

**GENERAL EVALUATION/REVISIONAL QUESTIONS**

1. The probabilities that Ade and Bayo passed a exam are 2/3 ad 3/5 respectively find the probability that (i) two of them passed (ii) two of them failed.

2. A six- sided die is thrown once. What is the probability that the result will be (a) an even number (b) a number less than 6 (c) a number greater than 2?

3. A universal set had 24 elements and A and B are subsets of the universal set such that n (A) = 14, n(B) =9 and n(A B) = 6. If P (a) is the probability of selecting an element belonging to set A, calculate (a) P (A) (b) P (A B) (c) P (B^{1}) (d) P(AU B)^{1}

Reading Assignment ; New Further Maths Project 2 page 198 – 210

**WEEKEND ASSIGNMENT**

1) In a single throw of a fair coin ,find the probability that a head appears a) ¾ b) ½ c) 2/3 d)¼

Two fair dice are tossed , find the probability that the total score is

2) prime number a) 5/12 b) ¾ c) ½ d) 5/36

3) less than 6 a) 5/36 b) 5/24 c) 5/6 d) 5/18

A bag contains 6 red and 4 blue identical marbles, if two marbles are picked one after the other without replacement, find the probability that both marbles are of

4) the same colour a) 8/15 b) 7/15 c) 1/3 d) 4/15

5) differentcolour a) 4/15 b) 7/15 c) 8/15 d) 4/9

**THEORY**

1) Two dice are thrown together, what is the probability of getting (i) at least 6 (ii) score greater than 8

2) A box contains 5 white and 3 black balls , if two balls are drawn one after the other with replacement what is the probability that both of them are (i) of the same colour (ii) of different colour

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