Question

Rework problem 9 from section 3.2 of your text, involving independent and disjoint events. For this problem, assume that Pr[A∪B]=0.7 and Pr[A]=0.25 .

Answer 1

Two events are said to be  Disjoint or Mutually Exclusive if the two events can not  happen at the same time.For example when we throw a die getting an even number is disjoint to getting an odd number.

I.e Probability(A∩B)=0

Let me explain this concept through venn diagram.

Pr[A∪B]=0.7, Pr[A]=0.25

Since events are Disjoint

Pr[A∩B]=0

Pr[A∪B]=Pr[A] + Pr[B]

0.7=0.25 +Pr[B]

0.7-0.25=Pr[B]

⇒Pr[B]=0.45=45/100=9/20

Now events are said to be independent if Pr[A and B]=Pr[A] ×Pr[B]

Events are said to be independent if occurrence of one is not affected by occurrence of other.For example getting multiple of 2 as one event and getting multiple of 3 as second event when we throw a die.

Pr[A∪B]=0.7, Pr[A]=0.25

Pr[A∪B]= Pr[A]+ Pr[B]-Pr[A∩B]

But Pr[A∩B]= Pr[A] ×Pr[B]

⇒Pr[A∪B]= Pr[A]+ Pr[B]- Pr[A] ×Pr[B]

⇒0.7=0.25+p-0.25×p

⇒0.7-0.25=p- 0.25 p

⇒0.45=0.75 p

⇒p= 0.45/0.75

⇒p =3/5