A certain type of toy-cars requires three AAA batteries, and the toy-cars will work only if all its batters have acceptable voltages. Assume that all the toy-cars contain batteries from the same supplier and that 90% of all batteries from this supplier have acceptable voltages.

1. What is the probability that a randomly selected toy-car is in working condition?

Now, in order to test the acceptance of the toy-cars from the same supplier, a quality control engineer randomly selects 10 toy-car from a lot. Suppose every selection are independent and X is a random variable showing the number of acceptable toy-cars out of 10.

2. What is the probability that among 10 randomly selected toy-cars that exactly half will work?
3. What is the probability that at least 2 toy-cars are functional out of 10?
4. What is the expected number of toy-cars that are acceptable? Interpret its meaning in the context of this problem.
5. What is standard deviation of toy-cars that are acceptable?

1. Suppose that the table below displays the results of a pregnancy test administered to a sample of 150 females that are either pregnant or not pregnant. Please answer the questions that follow based on the sample taken from a population. Note that every probability is an approximation of the population probabilities.
   1. What is the prevalence of pregnancy, i.e., P(S)?
   2. What is the probability that a randomly selected individual will be pregnant, i.e., P(Preg)?
   3. What is the probability that a randomly selected individual will test positive, i.e., P(Pos)?
   4. What is the number of females in the sample with a false positive test?
   5. What is the sensitivity of this test? That is, what is P(Pos|Preg)?
   6. What is the number of females in the sample with a false negative test?
   7. What is the specificity? That is, what is P(Neg|Preg^c)?
   8. What is the positive predictive value for this test, i.e., P(S|Pos)?
   9. What does the positive predictive value of this test mean contextually?

A survey of 2645 consumers by DDB Needham Worldwide of Chicago for public relations agency Porter/Novelli showed that how a company handles a crisis when at fault is one of the top influences in consumer buying decisions,with 73% claiming it is an influence. Quality of product was the number one influence, with 96% of consumers stating that quality influences their buying decisions. How a company handles complaints was number two, with 85% of consumers reporting it as an influence in their buying decisions. Suppose a random sample of 1,100 consumers is taken and each is asked which of these three factors influence their buying decisions. Appendix A Statistical Tables

a. What is the probability that more than 830 consumers claim that how a company handles a crisis when at fault is an influence in their buying decisions?

b. What is the probability that fewer than 1,030 consumers claim that quality of product is an influence in their buying decisions?

c. What is the probability that between 82% and 84% of consumers claim that how a company handles complaints is an influence in their buying decisions?

A fishing lake camp boasts that about 30% of the guests catch lake trout over 20 pounds on a 4-day fishing trip. Let n be a random variable that represents the first trip to the camp on which a guest catches a lake trout over 20 pounds.

(a) Write out a formula for the probability distribution of the random variable n.
P(n) =

(b) Find the probability that a guest catches a lake trout weighing at least 20 pounds for the first time on trip number 3. (Round your answer to three decimal places.)


(c) Find the probability that it takes more than three trips for a guest to catch a lake trout weighing at least 20 pounds. (Round your answer to three decimal places.)


(d) What is the expected number of fishing trips that must be taken to catch the first lake trout over 20 pounds? Hint: Use μ for the geometric distribution and round. (Round your answer to two decimal places.)
(    ) trips

The City Transit Authority plans to hire 14 new bus drivers. From a group of 100 qualified applicants, of whom 60 are men and 40 are women, 14 names are to be selected by lot. Suppose that Mary and John Lewis are among the 100 qualified applicants.

(a) What is the probability that Mary's name will be selected?

What is the probability that both Mary's and John's names will be selected? (Round your answer to four decimal places.)

(b) If it is stipulated that an equal number of men and women are to be selected (7 men from the group of 60 men and 7 women from the group of 40 women), what is the probability that Mary's name will be selected? (Round your answer to four decimal places.)

What is the probability that Mary's and John's names will be selected? (Round your answer to four decimal places.)

Beth and her husband have 8 good friends (inlcuding her two sisters) 5 of whom are men and 3 of whom are women. She won a radio contest with 6 national tickets. Her and her husband will definitely be attending the game but she would like to select friends to attend with her for the other tickets. She is having a hard time deciding, so she puts all 8 names in a hat and randomy draws the names.

a. Probability that she selects both of her sisters?

b. probability that she selects at least 2 men?

c. probability that she selects an equal number of men and women?

d. if she has to chang her methodology because two of the friends are married and wont attend seprately. What is the probability that she slects the married couple?

If possible, please type up or scan the page.

Suppose a box contains five coins (numbered 1, 2, 3, 4 and 5) but each coin has a different probability of obtaining a head when it is tossed. Let the p_i be given by 0,1/4,1/2,3/4 and 1.0, respectively for the coins, indexed by i=1,…,5.

1. Suppose a coin is randomly chosen from the box (i.e. each coin has the same probability of being chosen) and tossed. It shows a head. What are the posterior probabilities that the coin was number i?

2. If that coin were tossed again what is the probability of obtaining another head.

3. Suppose a tail had been obtained on the first toss, and the coin were tossed again. What is the probability a head would be obtained on the second toss?

Ten percent of the engines manufactured on an assembly line are defective. If engines are randomly selected one at a time and tested

a) What is the probability that the first non-defective engine will be found on the second trial?

b) What is the probability that the third non-defective engine will be found

i. on the fifth trial?

ii. on or before the fifth trial?

c. Find the mean and variance of the number(position) of the trial on which:

i. the first non-defective engine is found.

ii. the third non-defective engine is found.

d. Given that the first two engines tested were defective, what is the probability that at least two more engines must be tested before the first non-defective is found?

e. Find the probability that out of 8 randomly selected engines exactly 5 will be defective.

The English alphabet has 26 letters. There are 6 vowels. (a, e, i, o, u, and sometimes y). Suppose we randomly select 8 letters from the alphabet without replacement. Let X = the number of vowels chosen (including y as a vowel).

a. How many possible ways are there to select the 8 out of 26 letters (order does not matter) without replacement?
b. What is the probability that X = 2
c. What is the probability that X=1?
d. What is the probability that X < 3?
e. What is the expected value of X?
f. What is the standard deviation of X?
g. What is the probability that X is within 1.1 standard deviations of its expected value?
h Execute the R command "sample(letters,8)". How many vowels did you get?