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

Question

Allisa

2 months ago

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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 .

Mathematics

1 answer:

Svet_ta [12.7K] · Answer 1 · 2026-03-09T01:31:34+03:00

Events are considered Disjoint or Mutually Exclusive if they cannot occur simultaneously. For instance, rolling a die and landing on an even number is mutually exclusive of landing on an odd number.

In mathematical terms, Probability(A∩B)=0

Let’s illustrate this concept using a Venn diagram.

Pr[A∪B]=0.7, Pr[A]=0.25

Since the 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

Events are characterized as independent when Pr[A and B]=Pr[A] ×Pr[B]

They are independent if the occurrence of one event does not affect the likelihood of the occurrence of the other. For instance, when throwing a die, obtaining a multiple of 2 as one event and getting a multiple of 3 as another event.

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