You are given the following:

• Events [math]A[/math] and [math]B[/math] are independent
• [math]\operatorname{P}[A-B] + \operatorname{P}[B-A][/math] = 0.5
• [math]\operatorname{P}[A] [/math] = 0.65

Determine [math]\operatorname{P}[A \cap B][/math].

• 0.2275
• 0.3
• 0.325
• 1/3
• 0.5

An insurance company pays hospital claims. The number of claims that include emergency room or operating room charges is 85% of the total number of claims. The number of claims that do not include emergency room charges is 25% of the total number of claims. The occurrence of emergency room charges is independent of the occurrence of operating room charges on hospital claims.

Calculate the probability that a claim submitted to the insurance company includes operating room charges.

• 0.10
• 0.20
• 0.25
• 0.40
• 0.80

Let A, B, and C be events such that [math]\operatorname{P}[A] = 0.2[/math], [math]\operatorname{P}[B] = 0.1 [/math], and [math]\operatorname{P}[C] = 0.3 [/math]. The events A and B are independent, the events B and C are independent, and the events A and C are mutually exclusive.

Calculate [math]\operatorname{P}[A \cup B \cup C] . [/math]

• 0.496
• 0.540
• 0.544
• 0.550
• 0.600

Events [math]E[/math] and [math]F[/math] are independent. [math]\operatorname{P}[E] = 0.84[/math] and [math]\operatorname{P}[F] = 0.65[/math].

Calculate the probability that exactly one of the two events occurs.

• 0.056
• 0.398
• 0.546
• 0.650
• 0.944

In any 12-month period, the probability that a home is damaged by fire is 20% and the probability of a theft loss at a home is 30%. The occurrences of fire damage and theft loss are independent events.

Calculate the probability that a randomly selected home will either be damaged by fire or will have a theft loss, but not both, during the next year.

• 0.30
• 0.38
• 0.44
• 0.50
• 0.56

If [math]A [/math] is independent of itself, what are the possible values for [math]\operatorname{P}(A)[/math] ?

• 0 only
• 1 only
• 0 or 1 only
• 1/2 only
• Impossible to determine

Two fair dice, one red and one blue, are rolled.

Let A be the event that the number rolled on the red die is odd.

Let B be the event that the number rolled on the blue die is odd.

Let C be the event that the sum of the numbers rolled on the two dice is odd. Determine which of the following is true.

• A, B, and C are not mutually independent, but each pair is independent.
• A, B, and C are mutually independent.
• Exactly one pair of the three events is independent.
• Exactly two of the three pairs are independent.
• No pair of the three events is independent.

If [math]A[/math] and [math]B[/math] are independent, which of the following is always true?

• [math]A[/math] and [math]B^c[/math] are also independent
• [math]A^c[/math] and [math]B^c[/math] are not independent.
• [math]A^c[/math] and [math]B[/math] are not independent.
• [math]A\cap B[/math] is independent of [math]A[/math].
• [math]P(A\cup B) = P(A) + P(B)[/math].

Suppose the following holds:

• [math]A [/math] is independent of both [math]B [/math] and [math]C[/math]
• [math]\operatorname{P}(B \cup C ) = 0.7 [/math]
• [math]\operatorname{P}(A) = 0.5 [/math], [math]\operatorname{P}(B) = 0.5 [/math] and [math]\operatorname{P}(C) = 0.3 [/math]
• [math]\operatorname{P}(A \cup B \cup C) = 1[/math]

Determine [math]\operatorname{P}(A \cap B \cap C) [/math].

1. 0
2. 0.1
3. 0.15
4. 0.2
5. 0.3

A coin is tossed twice. Consider the following events.

[math]A[/math]: Heads on the first toss.

[math]B[/math]: Heads on the second toss.

[math]C[/math]: The two tosses come out the same.

Which one of the following statements is true?

• [math]A[/math], [math]B[/math], [math]C[/math] are independent.
• [math]C[/math] is independent of [math]A[/math] and [math]B[/math] but not of [math]A \cap B[/math].
• [math]C[/math] is mutually independent from [math]A[/math] but not of [math]B[/math].
• [math]C[/math] is mutually independent from [math]B[/math] and mutually independent of [math]A \cap B[/math].
• [math]C[/math] is independent of [math]A-B[/math] and also independent of [math]B-A[/math].

References

Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.