Q: Given the probability distribution for the number of men who have tinnitus among four randomly selected males: $$P(0) = 0.074$$, $$P(1) = 0.264$$, $$P(2) = 0.406$$, $$P(3) = 0.204$$, $$P(4) = 0.052. $$What is the expected number of men with tinnitus?

$$0 \times 0.074 + 1 \times 0.264 + 2 \times 0.406 + 3 \times 0.204 + 4 \times 0.052$$

Q: A bag contains 35 red, 10 white, and 5 blue marbles. If two marbles are drawn WITHOUT replacement, what is the probability that both are red?

$$\frac{35}{50} \times \frac{34}{49}$$

Question 1

In a U.S. Open tennis tournament, 839 player challenges to referee calls were recorded. If 200 challenges made by male players were accepted and 126 challenges made by female players were accepted, what is the probability that a randomly selected challenge was accepted by a female player?

Accepted Answer

$$\frac{126}{839}$$

Question 2

A bag contains 50 marbles: 35 are red, 10 are white, and 5 are blue. If two marbles are randomly selected WITH replacement, what is the probability that both are red?

Accepted Answer

$$\left(\frac{35}{50}\$$right)^2$$

Question 3

Assume that 40% of the population has brown eyes. What is the probability that among six randomly selected people, exactly four of them have brown eyes?

Accepted Answer

$$\binom{6}{4} (0.4)^4$ (0.6)^2$

Question 4

Given the probability distribution for the number of men who have tinnitus among four randomly selected males: $$P(0) = 0.074$$, $$P(1) = 0.264$$, $$P(2) = 0.406$$, $$P(3) = 0.204$$, $$P(4) = 0.052. $$What is the expected number of men with tinnitus?

Accepted Answer

$$0 	imes 0.074 + 1 	imes 0.264 + 2 	imes 0.406 + 3 	imes 0.204 + 4 	imes 0.052$$

Question 5

A statistics instructor finds that 70% of students arrive on time. If 8 students are randomly selected, what is the probability that exactly 6 students arrive on time?

Accepted Answer

$$\binom{8}{6} (0.7)^6$ (0.3)^2$

Question 6

If the mean red blood cell count of women is 4.577 and the standard deviation is 0.382, what is the z-score for a woman with a red blood cell count of 5.4?

Accepted Answer

$$z = \frac{5.4 - 4.577}{0.382}$$

Question 7

A poll finds that 77% of U.S. adults think political correctness is a big problem. If you randomly select 200 adults, what is the expected number who would say political correctness is a problem?

Accepted Answer

$$200 	imes 0.77$$

Question 8

If the probability that a randomly selected person does not use their cell phone once a day is 0.05, what is the probability that all 5 randomly selected cell phone owners do not use their cell phone once a day?

Accepted Answer

$$(0.05)^5$$

Question 9

Given a normal distribution with mean $$\mu = 0$$ and standard deviation $$\sigma = 1$$, what is the z-score for the 90th percentile?

Accepted Answer

$$z \approx 1.28$$

Question 10

A bone mineral density test score is normally distributed with mean 0 and standard deviation 1. What is the probability that a randomly selected subject has a score greater than 1?

Accepted Answer

$$P(z \u003e 1) = 1 - P(z \u003c 1) = 1 - 0.8413 = 0.1587$$

Question 11

If 25 women are randomly selected, what is the probability that the mean of their red blood cell counts is less than 4.444, given $$\mu = 4.577$$ and $$\sigma = 0.382$$?

Accepted Answer

$$P\left(\bar{x} \u003c 4.444\right)$$ where $$z = \frac{4.444 - 4.577}{0.382 / \sqrt{25}}$$

Question 12

A bag contains 35 red, 10 white, and 5 blue marbles. If two marbles are drawn WITHOUT replacement, what is the probability that both are red?

Accepted Answer

$$\frac{35}{50} 	imes \frac{34}{49}$$

Question 13

Given a normal distribution with mean $$\mu = 4.577$$ and standard deviation $$\sigma = 0.382$$, what percentage of women have red blood cell counts below 4.2?

Accepted Answer

$$P(z \u003c \frac{4.2 - 4.577}{0.382})$$

Question 14

If the probability that a randomly selected person does not use their cell phone once a day is 0.05, would it be 'unusual' if all 5 randomly selected cell phone owners do not use their cell phone once a day?

Accepted Answer

Yes, because $$(0.05)^5 is $$much less than 0.05, indicating a rare event.

Question 15

A poll result was obtained by Internet users logging on to a website and deciding whether to respond. What type of sample is this, and what does it suggest about the validity of the results?

Accepted Answer

It is a voluntary response sample, which may not be representative and can introduce bias.

Question 16

Given a standard normal distribution, what is the z-score for $$P_{25} ($$the 25th percentile)?

Accepted Answer

$$z \approx -0.674$$

Question 17

Four males with an X-linked genetic disorder each have one child. The random variable $$x is $$the number of children among the four who inherit the disorder. What is the expected value $$E(x)$$ given the probability distribution: $$P(0) = 0.0625$$, $$P(1) = 0.25$$, $$P(2) = 0.375$$, $$P(3) = 0.25$$, $$P(4) = 0.0625$$?

Accepted Answer

$$0 	imes 0.0625 + 1 	imes 0.25 + 2 	imes 0.375 + 3 	imes 0.25 + 4 	imes 0.0625$$

Question 18

If a bone mineral density test score is normally distributed with mean 0 and standard deviation 1, what is the probability that a randomly selected subject has a score less than 0.5?

Accepted Answer

$$P(z \u003c 0.5) = 0.6915$$

Probability, Distributions, and Normal Curve Applications in Statistics

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