Question 1

Which of the following statements best describes a discrete random variable?

Accepted Answer

A variable whose values are finite or countable, such as the number of heads in coin tosses.

Question 2

Which of the following is NOT a requirement for a probability distribution?

Accepted Answer

The random variable must be categorical.

Question 3

Given the probability distribution for the number of females in two births: $$x = 0, 1, 2$$ with $$P(0) = 0.25$$, $$P(1) = 0.5$$, $$P(2) = 0.25$$, what is the mean ($$\mu) of $$this distribution?

Accepted Answer

$$\mu = 1.0$$

Question 4

Which formula should be used to calculate the variance ($$\$$sigma^2$$) of a discrete probability distribution for manual calculations?

Accepted Answer

$$\$$sigma^2$$ = \sum[x^2$ \cdot P(x)] - \mu^2$

Question 5

If the mean ($$\mu) of a $$probability distribution is 1.0 and the standard deviation ($$\sigma) is 0.7$$, according to the range rule of thumb, which value would be considered significantly high?

Accepted Answer

Any value greater than or equal to $$\mu + 2\sigma = 2.4$$

Question 6

A binomial probability distribution requires all of the following EXCEPT:

Accepted Answer

Each trial must have more than two possible outcomes.

Question 7

In a binomial experiment with $$n = 10$$ trials and probability of success $$p = 0.05$$, what is the probability of exactly 2 successes? Use the binomial formula: $$P(x) = \frac{n!}{x!(n-x)!} p^x q$$^{n-x}$$ where q = 1 - p$.

Accepted Answer

$$P(2) = \frac{10!}{2!8!} (0.05)^2$ (0.95)^8$

Question 8

Which scenario violates the requirements for a binomial probability distribution?

Accepted Answer

Sampling without replacement from a small population.

Question 9

For a binomial distribution with $$n = 460$$ and $$p = 0.5$$, what is the mean ($$\mu) $$and standard deviation ($$\sigma$$)?

Accepted Answer

$$\mu = 230$$, $$\sigma = \sqrt{115}$$

Question 10

If the probability of getting 252 or more heads in 460 coin tosses is $$0.0224$$, what can be concluded about this result?

Accepted Answer

252 heads is a significantly high number of heads.

Question 11

In the context of expected value, which formula is used to calculate the expected value ($$E) of a $$discrete random variable?

Accepted Answer

$$E = \sum[x \cdot P(x)]$$

Question 12

You bet $$5 on the number 7 in roulette. The probability of losing 5 is$$ $$\frac{37}{38}$$ and the probability of winning $$175 is $$\frac{1}{38}$. What is the expected value of your bet?

Accepted Answer

$$E = (-5) \cdot \frac{37}{38} + 175 \cdot \frac{1}{38}$$

Question 13

Which of the following best explains why we use '0+' in probability tables?

Accepted Answer

To indicate a probability value that is positive but very small, so rounding to 0 would be misleading.

Question 14

If you are sampling without replacement from a population, under what condition can you treat the events as independent for binomial calculations?

Accepted Answer

If the sample size is no more than 5% of the population size.

Question 15

A casino game offers a net gain of $$5 with probability $$\frac{244}{495}$$ and a net loss of 5 $$with probability $$\frac{251}{495}. $$What is the expected value of a $5 bet?

Accepted Answer

$$E = 5 \cdot \frac{244}{495} + (-5) \cdot \frac{251}{495}$$

Question 16

For a binomial distribution, which formula gives the standard deviation ($$\sigma$$)?

Accepted Answer

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

Question 17

In a binomial experiment with $$n = 5$$ and $$p = 0.05$$, what is the probability of fewer than 3 successes?

Accepted Answer

$$P(0) + P(1) + P(2)$$

Question 18

Which of the following best describes the use of the range rule of thumb in identifying significant values in a probability distribution?

Accepted Answer

Values outside $$\mu \pm 2\sigma$$ are considered significant.

Question 19

In decision theory, why is expected value important when comparing different bets?

Accepted Answer

It helps determine which bet is better in the long run, regardless of the size of individual payoffs.

Chapter 5.2

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