Question

(SHOW ALL CALCULATIONS PLEASE) [40]

1. Define/Illustrate Mutual Exclusivity and Independence. (2)

2. A,B,C ⊆U. n(A∩B)=9,n(A∩C)=5,
n(B∩C)=7. n(A∩B^c∩C^c )=24,
n(B∩A^c∩C^c )=7,n(C∩A^c∩B^c )=6.
n(A)=n(B)+n(C) .
Determine:
(a) n(A∩B∩C) (5)
(b) n(U) if A,B and C are exhaustive.

Answer 1

1. Two events are said to be mutually exclusive, when their occurrence is not simultaneous.

Occurrence of one event will result in the non-occurrence of the other.

Let $\empty$ denote the null event (the event consisting of no outcomes whatsoever).

When $A\cap B=\empty$ , $A$ and $B$ are said to be mutually exclusive or disjoint events.

$$P(A\cap B)=0$$

Two events are said to be independent, when the occurrence of one event cannot control the occurrence of other.

Occurrence of one event will have no influence on the occurrence of the other.

$$P(A\cap B)=P(A)P(B)$$

2.

(a) Let $n(A\cap B \cap C)=x$

Given $n(A)=n(B)+n(C).$

Then

$$24+9-x+5-x+x=7+9-x+7-x+x$$

$$+6+5-x+7-x+x$$

$$38-x=41-2x$$

$$x=3$$

$n(A\cap B \cap C)=3$

(b)

$$n(U)=24+6+7+5-x+7-x+9-x+x$$

$$=58-2x$$

If $x=3$

$$n(U)=58-2(3)=52$$