Q: Suppose you toss a fair coin 100 times. What is the expected number of heads?

$$E(X) = 100 \times \frac{1}{2} = 50$$

Q: If $$n = 100$$ and $$p = 0.5 in a $$binomial experiment, what is the standard deviation?

$$\sqrt{100 \times 0.5 \times 0.5} = \sqrt{25} = 5$$

Q: If you toss a coin 1000 times, what is the expected number of tails?

$$E(X) = 1000 \times \frac{1}{2} = 500$$

Question 1

Which of the following best describes a random variable (RV) in the context of a coin toss experiment?

Accepted Answer

A numerical measure of the outcome determined by chance.

Question 2

In a binomial experiment, which of the following conditions must be satisfied?

Accepted Answer

The probability of success must be the same for each trial.

Question 3

Suppose you toss a fair coin 100 times. What is the expected number of heads?

Accepted Answer

$$E(X) = 100 	imes \frac{1}{2} = 50$$

Question 4

Which formula correctly represents the mean (expected value) of a discrete probability distribution?

Accepted Answer

$$\sum x_i P(x_i)$$

Question 5

A random variable is said to be discrete if:

Accepted Answer

It can take on a countable number of values.

Question 6

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

Accepted Answer

The probability of success must change from trial to trial.

Question 7

If $$n = 100$$ and $$p = 0.5 in a $$binomial experiment, what is the standard deviation?

Accepted Answer

$$\sqrt{100 	imes 0.5 	imes 0.5} = \sqrt{25} = 5$$

Question 8

Which statement best explains why a coin toss experiment is considered a binomial experiment?

Accepted Answer

Each toss has only two possible outcomes, and the probability of each outcome remains constant.

Question 9

Given a probability distribution where $$P(X)$$ represents the probability of outcome $$X$$, which of the following must be true?

Accepted Answer

$$\sum P(X) = 1$$

Question 10

In a binomial experiment with $$n$$ trials and probability of success $$p$$, what is the formula for the mean (expected value)?

Accepted Answer

$$\mu = n p$$

Question 11

Which formula gives the standard deviation of a binomial random variable?

Accepted Answer

$$\sigma = \sqrt{n p (1-p)}$$

Question 12

A probability distribution is valid if:

Accepted Answer

All probabilities are between 0 and 1, and their sum is 1.

Question 13

If you toss a coin 1000 times, what is the expected number of tails?

Accepted Answer

$$E(X) = 1000 	imes \frac{1}{2} = 500$$

Question 14

Which of the following best describes a continuous random variable?

Accepted Answer

It can take on any value within a given interval.

Question 15

In a probability distribution, what does the value $$x$$ represent?

Accepted Answer

A possible outcome of the random variable.

Question 16

If $$X is a $$random variable representing the number of heads in 100 coin tosses, what is the probability of getting exactly 50 heads (assuming a fair coin)?

Accepted Answer

$$P(X=50) = \binom{100}{50} \left(\frac{1}{2}\$$right)^{100}$$

Question 17

Which of the following is an example of a discrete random variable?

Accepted Answer

The number of students present in a classroom.

Question 18

If the probability of success in a binomial experiment is $$p$$, what is the probability of failure?

Accepted Answer

$$1 - p$$

Question 19

Which of the following best describes the Empirical Rule?

Accepted Answer

It states that for a normal distribution, about 68% of values fall within one standard deviation of the mean.

Question 20

In a binomial experiment, if the probability of success is $$0.3$$ and the probability of failure is $$0.7$$, what is the expected number of successes in 200 trials?

Accepted Answer

$$E(X) = 200 	imes 0.3 = 60$$

Chapter 6 - Discrete Probability Distributions and Binomial Experiments

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