Consider randomly selecting a student at a certain university, and let A denote the event that...

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Consider randomly selecting a student at a certain university, and let A denote the event that...

Consider randomly selecting a student at a certain university, and let A denote the event that the selected individual has a Visa credit card and B be the analogous event for a MasterCard where P(A) = 0.45, P(B) = 0.35, and P(A ❩ B) = 0.30. Calculate and interpret each of the following probabilities (a Venn diagram might help). (Round your answers to four decimal places.)

(a)    P(B | A)

(b)    P(B' | A)

(c)    P(A | B)

(d)    P(A' | B)

(e) Given that the selected individual has at least one card, what is the probability that he or she has a Visa Card?

math Statistics-And-Probability

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Answer 1

Answer #1

Concepts and reason

The probability of the event E is denoted as .

The probability value will be in the range 0 to 1. The opposite or complement of an event E is the event (not E). It is denoted as .

The probability that an event will occur given that some other event has already happened is called conditional probability.

Fundamentals

The probability of complement of an event E is,

P(E)=1- P(E)

The conditional probability for any two events is given by,

P(4|B)=P(AMB)
P(B)

Properties:

1. P(A|B) =1-P(AB)

2. P(B| A) =1-P(BA)

3. P(B|4)=P(AB)

4. P(AUB)=P(A) + P(B)- P(
A
B
)

5. P(A|AUB)=
PAn(AUB))
P(AUB)

The Venn diagram is given by,

AB AB
AB

(a)

Let A be the event that the selected individual has a Visa credit card and B be the analogous event for a MasterCard.

From the given information,

P(A)=0.45,P(B)=0.35, and P(An B)=0.30 .

The Venn diagram is given by,

0.15
(0.30)
0.05

Compute the probability value of P(BA) .

P(BIA)- P(An B)
P(A)
0.30
0.45
= 0.6667

Therefore, the probability value is 0.6667.

(b)

Compute the probability value of P(
BA) .

P(B| A)=1-P(BA)
= 1-0.6667
= 0.3333

Therefore, the probability value is 0.3333.

(c)

Let A be the event that the selected individual has a Visa credit card and B be the analogous event for a MasterCard.

From the given information,

.

Compute the probability value of P(AB) .

P(AB)=P(An B)
P(B)
0.30
0.35
= 0.8571

Therefore, the probability value is 0.8571.

(d)

Compute the probability value of P(A| B) .

P(A| B)=1-P(AB)
= 1-0.8571
= 0.1429

Therefore, the probability value is 0.1429.

(e)

Let A be the event that the selected individual has a Visa credit card and B be the analogous event for a MasterCard.

From the given information,

.

Compute the probability of having visa card given selected individual has at least one card.

P(A|AUB)=
P(AN(AUB))
P(AUB)
P(A)
P(AUB)
P(A)
P(A) + P(B)- P(An B)
0.45
0.45 +0.35 -0.30
= 0.9

Therefore, the probability of having visa card given selected individual has at least one card is 0.9.

Ans: Part a

The probability value is 0.6667.

Part b

The probability value is 0.3333.

Part c

The probability value is 0.8571.

Part d

The probability value is 0.1429.

Part e

The probability of having visa card given selected individual has at least one card is 0.9.

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