Events A and B are mutually exclusive, with P(A) = 0.25 and P(B) = 0.40. Event C is indepe… — GRE General Test

Events $A$ and $B$ are mutually exclusive, with $P(A) = 0.25$ and $P(B) = 0.40$ . Event $C$ is independent of both event $A$ and event $B$ , with $P(C) = 0.50$ . What is the probability that event $C$ occurs and at least one of events $A$ or $B$ occurs?

A
$0.050$
B
$0.125$
$0.325$ Cevap
D
$0.650$
E
$0.825$

Cevap

The probability that event

C

occurs and at least one of events

A

B

occurs is

0.325

The correct answer is

0.325

. First, since

A

and

B

are mutually exclusive events, the probability of at least one of them occurring is

P(A \text{ or } B) = P(A) + P(B) = 0.25 + 0.40 = 0.65

. Second, because event

C

is independent of both events, the probability that

C

occurs AND at least one of

A

B

occurs is given by the multiplication rule for independent events:

P(C) \times P(A \text{ or } B) = 0.50 \times 0.65 = 0.325

Adım Adım Çözüm

Calculate the probability of the union of mutually exclusive events

A

and

B

P(A \text{ or } B) = P(A) + P(B) = 0.25 + 0.40 = 0.65

Since

A

and

B

are mutually exclusive,

P(A \cap B) = 0

, so their combined probability is simply the sum of their individual probabilities.

Calculate the joint probability of event

C

and event

(A \text{ or } B)

P(C \text{ and } (A \text{ or } B)) = P(C) \times P(A \text{ or } B) = 0.50 \times 0.65 = 0.325

Event

C

is independent of both

A

and

B

, which implies

C

is independent of

(A \text{ or } B)

. Therefore, the joint probability is found by multiplying their individual probabilities.

Anahtar Kavram

Probability rules for mutually exclusive events (addition rule) and independent events (multiplication rule).

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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