Q: Which formula correctly expresses the formal addition rule for two events $$A$$ and $$B$$?

$$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$$

Q: Suppose you randomly select two subjects from a group of 50, where 45 have positive test results and 5 have negative test results. If you select with replacement, what is the probability that the first is positive and the second is negative?

$$\dfrac{45}{50} \times \dfrac{5}{50} = 0.09$$

Q: If two events $$A$$ and $$B$$ are independent, which of the following is true?

$$P(A \text{ and } B) = P(A) \times P(B)$$

Q: In the Airbus 310 example, the probability that a single hydraulic system fails is $$0.002. $$What is the probability that all three independent hydraulic systems fail on a flight?

$$0.002 \times 0.002 \times 0.002 = 0.000000008$$

Question 1

Which of the following statements best describes disjoint (mutually exclusive) events?

Accepted Answer

Disjoint events are events that cannot occur at the same time.

Question 2

Suppose $$P(A) = 0.3$$, $$P(B) = 0.5$$, and $$P(A 	ext{ and } B) = 0.1. $$What is $$P(A 	ext{ or } B)$$?

Accepted Answer

$$0.3 + 0.5 - 0.1 = 0.7$$

Question 3

If the probability of a randomly selected person having sleepwalked is $$0.292$$, what is the probability that a randomly selected person has NOT sleepwalked?

Accepted Answer

$$1 - 0.292 = 0.708$$

Question 4

Which formula correctly expresses the formal addition rule for two events $$A$$ and $$B$$?

Accepted Answer

$$P(A 	ext{ or } B) = P(A) + P(B) - P(A 	ext{ and } B)$$

Question 5

A clinical trial randomly selects a participant who is either male or female. What is the relationship between the events 'selecting a male' and 'selecting a female'?

Accepted Answer

The events are disjoint (mutually exclusive).

Question 6

Suppose you randomly select two subjects from a group of 50, where 45 have positive test results and 5 have negative test results. If you select with replacement, what is the probability that the first is positive and the second is negative?

Accepted Answer

$$\dfrac{45}{50} 	imes \dfrac{5}{50} = 0.09$$

Question 7

If two events $$A$$ and $$B$$ are independent, which of the following is true?

Accepted Answer

$$P(A 	ext{ and } B) = P(A) 	imes P(B)$$

Question 8

A student is randomly selected from a university. Event A: the student is taking a statistics course. Event B: the student is female. Are these events disjoint?

Accepted Answer

No, because a student can be both taking statistics and be female.

Question 9

If $$P(A) = 0.4$$, what is $$P(A')$$, the probability that event $$A$$ does NOT occur?

Accepted Answer

$$1 - 0.4 = 0.6$$

Question 10

In the Airbus 310 example, the probability that a single hydraulic system fails is $$0.002. $$What is the probability that all three independent hydraulic systems fail on a flight?

Accepted Answer

$$0.002 	imes 0.002 	imes 0.002 = 0.000000008$$

Question 11

A system has three independent backup components, each with a probability of failure $$0.002. $$What is the probability that at least one system works (i.e., not all fail)?

Accepted Answer

$$1 - 0.000000008 = 0.999999992$$

Question 12

If you randomly select three adults from a population where 10% use drugs, and the population is very large, what is the probability that all three use drugs?

Accepted Answer

$$(0.10)^3 = 0.001$

Question 13

Which of the following best describes the principle of redundancy in engineering systems?

Accepted Answer

Redundancy means duplicating critical components to decrease the risk of total system failure.

Question 14

When sampling without replacement from a population, under what condition can you treat the selections as independent?

Accepted Answer

When the sample size is less than 5% of the population size.

Question 15

Which of the following is a compound event?

Accepted Answer

Selecting a card from a deck and rolling a die.

Question 16

If $$P(A) = 0.5$$ and $$P(B|A) = 0.2$$, what is $$P(A 	ext{ and } B)$$ using the multiplication rule?

Accepted Answer

$$0.5 	imes 0.2 = 0.1$$

Question 17

Which word is associated with the addition rule in probability?

Accepted Answer

"or"

Question 18

Which word is associated with the multiplication rule in probability?

Accepted Answer

"and"

Question 19

If $$P(A) = 0.7$$ and $$P(B) = 0.2$$, and $$A$$ and $$B$$ are disjoint, what is $$P(A 	ext{ or } B)$$?

Accepted Answer

$$0.7 + 0.2 = 0.9$$

Addition Rule and Multiplication Rule in Probability

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