Question 1

A fair six-sided die is rolled 60 times. According to the Law of Large Numbers, what is most likely to happen as the number of rolls increases?

Accepted Answer

The proportion of each outcome will get closer to $$\frac{1}{6}.$$

Question 2

Which of the following is an example of a subjective probability?

Accepted Answer

A doctor estimates a 70% chance of recovery for a patient based on experience.

Question 3

If $$A$$ and $$B$$ are mutually exclusive events, which of the following is true?

Accepted Answer

$$P(A \cap B) = 0$$

Question 4

A bag contains 5 red and 7 blue marbles. If one marble is drawn at random, what is the probability it is not blue?

Accepted Answer

$$\frac{5}{12}$$

Question 5

Which of the following statements about independent events is correct?

Accepted Answer

If $$A$$ and $$B$$ are independent, then $$P(A \cap B) = P(A) 	imes P(B).$$

Question 6

A survey found that 60% of people like coffee, 50% like tea, and 30% like both. What is the probability that a randomly selected person likes coffee or tea?

Accepted Answer

$$0.80$$

Question 7

Given $$P(A) = 0.4$$, $$P(B) = 0.5$$, and $$P(A \cap B) = 0.2$$, what is $$P(A|B)$$?

Accepted Answer

$$0.4$$

Question 8

Which of the following is a discrete random variable?

Accepted Answer

The number of defective light bulbs in a box of 100.

Question 9

A probability distribution for a discrete random variable $$X is $$valid if:

Accepted Answer

All probabilities are between 0 and 1, and the sum of all probabilities is 1.

Question 10

If $$X is a $$discrete random variable with the following probability distribution: $$P(X=0)=0.2$$, $$P(X=1)=0.5$$, $$P(X=2)=0.3$$, what is the expected value $$E[X]$$?

Accepted Answer

$$1.1$$

Question 11

Which of the following is NOT a requirement for a binomial experiment?

Accepted Answer

The probability of success changes from trial to trial.

Question 12

A fair coin is flipped 10 times. What is the probability of getting exactly 4 heads? (Use the binomial formula: $$P(X=k) = \binom{n}{k} p^k (1-p$$)^{n-k}$$)

Accepted Answer

$$\binom{10}{4} (0.5)^4$ (0.5)^6$

Question 13

For a binomial random variable with $$n=20$$ and $$p=0.3$$, what is the mean $$\mu$$?

Accepted Answer

$$6$$

Question 14

A medical test for a disease has a 95% chance of correctly identifying a diseased person (sensitivity) and a 90% chance of correctly identifying a healthy person (specificity). If 1% of the population has the disease, what is the probability that a randomly selected person tests positive? (Use the law of total probability.)

Accepted Answer

$$0.0595$$

Question 15

Which of the following situations is best modeled by a binomial distribution?

Accepted Answer

Counting the number of defective products in a batch of 50, where each product is either defective or not.

Question 16

If $$P(A) = 0.3$$, $$P(B) = 0.4$$, and $$A$$ and $$B$$ are independent, what is $$P(A \cap B)$$?

Accepted Answer

$$0.12$$

Question 17

A two-way (contingency) table shows that 40 students study math, 30 study science, and 10 study both. What is the probability that a randomly selected student studies math or science?

Accepted Answer

$$0.6$$

Question 18

When can the normal approximation to the binomial distribution be used?

Accepted Answer

When $$np \geq 10$$ and $$n(1-p) \geq 10$$

Question 19

A tree diagram is most useful for:

Accepted Answer

Visualizing all possible outcomes and their probabilities in multi-stage experiments.

Question 20

If $$P(A|B) = 0.7$$ and $$P(B) = 0.5$$, what is $$P(A \cap B)$$?

Accepted Answer

$$0.35$$

Probability and Random Variables: Structured Study Notes

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