Q: A binomial experiment consists of 10 trials, each with a probability of success $$p = 0.3. $$What is the expected number of successes?

$$E(X) = np = 10 \times 0.3 = 3$$

Q: A moderate seismic activity occurs with probability $$0.08 in $$Japan at a given time. If you invest $$1000$$ and lose $$500 if it $$occurs, what is your expected loss?

$$E = 0.08 \times (-500) + 0.92 \times 0 = -40$$

Q: A fair coin is tossed 4 times. What is the probability of getting exactly 2 heads?

$$P(X=2) = \binom{4}{2} (0.5)^2$ (0.5)^2$ = 6 \times 0.25 \times 0.25 = 0.375$$

Q: A medical test for a rare disease has a 98% accuracy rate. If 200 people are tested and 1% actually have the disease, what is the expected number of true positives?

$$E = 200 \times 0.01 \times 0.98 = 1.96$$

Question 1

A binomial experiment consists of 10 trials, each with a probability of success $$p = 0.3. $$What is the expected number of successes?

Accepted Answer

$$E(X) = np = 10 	imes 0.3 = 3$$

Question 2

Which of the following is NOT a characteristic of a binomial experiment?

Accepted Answer

The probability of success changes from trial to trial.

Question 3

Suppose you toss a fair six-sided die twice. Let $$X be $$the number of faces that show an even number. What is the probability that $$X = 2$$?

Accepted Answer

$$P(X=2) = \frac{9}{36} = 0.25$$

Question 4

A moderate seismic activity occurs with probability $$0.08 in $$Japan at a given time. If you invest $$1000$$ and lose $$500 if it $$occurs, what is your expected loss?

Accepted Answer

$$E = 0.08 	imes (-500) + 0.92 	imes 0 = -40$$

Question 5

Which formula correctly calculates the standard deviation of a binomial distribution?

Accepted Answer

$$\sigma = \sqrt{npq}$$

Question 6

In a class, 70% of students pass the final exam. If 5 students are randomly selected, what is the probability that exactly 3 pass?

Accepted Answer

$$P(X=3) = \binom{5}{3} (0.7)^3$ (0.3)^2$

Question 7

If the mean of a discrete random variable $$X is$$ $$\mu = 10.71$$ and the standard deviation is $$\sigma = 7.8166$$, what is the variance?

Accepted Answer

$$\$$sigma^2$$ = (7.8166)^2$ = 61.11$$

Question 8

A fair coin is tossed 4 times. What is the probability of getting exactly 2 heads?

Accepted Answer

$$P(X=2) = \binom{4}{2} (0.5)^2$ (0.5)^2$ = 6 	imes 0.25 	imes 0.25 = 0.375$$

Question 9

Which of the following best describes the sample space for tossing two fair six-sided dice?

Accepted Answer

There are 36 possible ordered pairs.

Question 10

If $$X is a $$binomial random variable with $$n = 20$$ and $$p = 0.05$$, what is the probability of getting zero successes?

Accepted Answer

$$P(X=0) = (1-0.05$$)^{20}$$ = 0.95$$^{20}$$

Question 11

A medical test for a rare disease has a 98% accuracy rate. If 200 people are tested and 1% actually have the disease, what is the expected number of true positives?

Accepted Answer

$$E = 200 	imes 0.01 	imes 0.98 = 1.96$$

Question 12

Which scenario is best modeled by a binomial distribution?

Accepted Answer

Counting the number of defective bulbs in a batch of 100, where each bulb has a fixed probability of being defective.

Question 13

If the probability of a student withdrawing from a physics course is $$0.3$$, and 7 students are enrolled, what is the probability that exactly 2 withdraw?

Accepted Answer

$$P(X=2) = \binom{7}{2} (0.3)^2$ (0.7)^5$

Question 14

A random variable $$X$$ has the following probability distribution: $$P(X=0)=0.2$$, $$P(X=1)=0.5$$, $$P(X=2)=0.3. $$What is the expected value of $$X$$?

Accepted Answer

$$E(X) = 0 	imes 0.2 + 1 	imes 0.5 + 2 	imes 0.3 = 1.1$$

Question 15

Which of the following is the correct formula for the probability of exactly $$k$$ successes in $$n$$ binomial trials?

Accepted Answer

$$P(X=k) = \binom{n}{k} p^k (1-p$$)^{n-k}$$

Question 16

A game has a 20% chance of winning $$100$$ and an 80% chance of losing $$50. $$What is the expected value of playing the game once?

Accepted Answer

$$E = 0.2 	imes 100 + 0.8 	imes (-50) = 20 - 40 = -20$$

Question 17

If a binomial distribution has $$n = 12$$ and $$p = 0.6$$, what is the standard deviation?

Accepted Answer

$$\sigma = \sqrt{12 	imes 0.6 	imes 0.4} = \sqrt{2.88} \approx 1.697$$

Question 18

A student claims that the probability of getting at least one six in two tosses of a fair six-sided die is $$\frac{11}{36}. Is $$this correct?

Accepted Answer

No, the correct probability is $$1 - (\frac{5}{6})^2$ = 1 - \frac{25}{36} = \frac{11}{36}$$

Discrete Probability Distributions and Binomial Distribution

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