Question 1

Given the table below:

Wore Helmet | Head Injuries: 48 | NO Head Injuries: 656
Did NOT Wear Helmet | Head Injuries: 480 | NO Head Injuries: 2330

If one person is randomly selected, what is the probability that the person did NOT wear a helmet and was NOT injured?

$$P(	ext{Did NOT Wear Helmet and NO Head Injury}) = ?$$

Accepted Answer

$$\frac{2330}{3514}$$

Question 2

Using the same table, what is the probability that a randomly selected person had a head injury, given that they did NOT wear a helmet?

$$P(	ext{Head Injury} \,|\, 	ext{Did NOT Wear Helmet}) = ?$$

Accepted Answer

$$\frac{480}{2810}$$

Question 3

Suppose that eye color in the United States is distributed as 40% brown, 35% blue, 12% green, 7% gray, and 6% hazel. If two people are randomly selected, what is the probability that at least one of them has brown eyes?

$$P(	ext{At least one brown}) = ?$$

Accepted Answer

$$1 - (0.6)^2$ = 0.64$$

Question 4

A Deal or No Deal contestant has to choose between an offer of $$193,000 or continuing the game. The five equally likely amounts left in play are 75$$, $$305, 570$$,000, $$500,000, and 1$$,000,000. What is the expected value of her winnings if she continues the game?

$$E = \frac{75 + 305 + 570,000 + 500,000 + 1,000,000}{5}$$

Accepted Answer

$$414,476$$

Question 5

Suppose that 'Inability to get along with others' is the reason cited in 17% of worker firings. In a sample of 5 firings, what is the probability that at least four of those employees were fired due to inability to get along with others?

$$P(X \geq 4)$$ where $$X \sim 	ext{Binomial}(n=5, p=0.17)$$

Accepted Answer

$$P(X=4) + P(X=5) = {5 \choose 4}(0.17)^4$(0.83)^1$$ + {5 \choose 5}$$(0.17)^5$

Question 6

A city has 535 homes in the combined area of 576 subdivisions. What is the mean number of homes per subdivision?

$$	ext{Mean} = \frac{535}{576}$$

Accepted Answer

$$0.93$$

Question 7

If a subdivision is randomly selected, what is the probability that it had no homes?

$$P(	ext{No homes}) = \frac{576 - 535}{576}$$

Accepted Answer

$$\frac{41}{576}$$

Question 8

How many regions would you expect to have no homes if the city has 229 subdivisions? Use the probability from the previous question.

$$E = 229 	imes \frac{41}{576}$$

Accepted Answer

$$16.3$$

Question 9

Assume that women's heights are normally distributed with a mean of 66.3 inches and a standard deviation of 2.5 inches. What is the $$z-$$score for a woman whose height is 71.5 inches?

$$z = \frac{71.5 - 66.3}{2.5}$$

Accepted Answer

$$2.08$$

Question 10

Using the same normal distribution, what percentage of women have heights that exceed 71.5 inches? (Use the $$z-$$score from the previous question and standard normal tables.)

Accepted Answer

$$1.88\%$$

Question 11

What is the probability that a randomly selected woman has a height between 66.5 and 71.5 inches? (Mean $$=66.3$$, SD $$=2.5)$$

Accepted Answer

$$P(0.08 \u003c z \u003c 2.08)$$

Question 12

If two people are randomly selected from the helmet study, what is the probability that both wore helmets?

$$P(	ext{Both wore helmets}) = \frac{704}{3514} 	imes \frac{703}{3513}$$

Accepted Answer

$$0.0403$$

Question 13

In the helmet study, what is the probability that a randomly selected person was injured, given that they wore a helmet?

$$P(	ext{Head Injury} \,|\, 	ext{Wore Helmet}) = \frac{48}{704}$$

Accepted Answer

$$0.068$$

Question 14

Suppose the probability that a person has blue eyes is 35%. If three people are randomly selected, what is the probability that all three have blue eyes?

$$P = (0.35)^3$

Accepted Answer

$$0.0429$$

Question 15

In the Deal or No Deal scenario, if the contestant accepts the $$193,000 offer, what is the opportunity cost?

$$	ext{Opportunity Cost} = 	ext{Expected Value} - 	ext{Offer}$

Accepted Answer

$$221,476$$

Question 16

If the probability that a randomly selected subdivision has no homes is $$\frac{41}{576}$$, what is the expected number of subdivisions with no homes in a city of 1000 subdivisions?

Accepted Answer

$$71.2$$

Question 17

If the probability that a person is fired for inability to get along with others is 0.17, what is the probability that in a sample of 5 firings, exactly 2 were for this reason?

$$P(X=2) = {5 \choose 2}(0.17)^2$(0.83)^3$

Accepted Answer

$$0.170$$

Question 18

If the heights of women are normally distributed with mean $$66.3$$ and SD $$2.5$$, what is the probability that a randomly selected woman is shorter than $$63.8$$ inches?

$$z = \frac{63.8 - 66.3}{2.5}$$

Accepted Answer

$$0.1587$$

Statistics Study Notes: Probability, Conditional Probability, and Normal Distribution

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