1) What is the probability that 4 randomly selected people ...



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Practice Exam #2

Math 5

Spring 1998

1) What is the probability that 4 randomly selected people all have different birthdays?

2) Choose all that apply. Describe the differences in the Poisson and the binomial

distribution.

a) The Poisson computes probabilities for occurrences of events over some

interval.

b) The Poisson distribution is affected only by the mean, whereas the binomial is

affected by sample size n and probability p.

c) The binomial is discrete and the Poisson is continuous

d) There are no differences

e) The Poisson distribution has discrete values from 1, 2, 3, . . . with no upper

limit. A binomial distribution has discrete values from 1, 2, 3, to n; that is, the

upper limit of values is n.

3) A study of consumer smoking habits includes 156 people in the 18-22 age bracket (58

of whom smoke), 123 in the 23-30 age bracket (37 of whom smoke), and 100 in the

31-40 age bracket (27 of whom smoke).

a) Create a contingency table for the information given.

b) Find the probability of getting someone who is age 23-30 or smokes.

c) Find the probability of getting a smoker that is 23-30.

d) Find the probability of getting someone who is 18-22 or 23-30.

e) Find the probability of getting someone who doesn't smoke and is 31-40

4) Sammy and Sally each carry a bag containing a banana, a chocolate bar, and a licorice

stick. Simultaneously, they take out a single food item and consume it. The possible

pairs of food items that Sally and Sammy consumed are as follows.

chocolate bar - chocolate bar

licorice stick - chocolate bar

banana - banana

chocolate bar - licorice stick

licorice stick - licorice stick

chocolate bar - banana

banana - licorice stick

licorice stick - banana

banana - chocolate bar

a) Find the probability that exactly one licorice stick was eaten.

b) Find the probability that at least one banana was eaten.

5) Determine whether the following is a probability distribution. If not, identify the

requirement that is not satisfied.

|x |P(x) |

|0 |0.191 |

|1 |0.231 |

|2 |-0.014 |

|3 |0.157 |

|4 |0.268 |

|5 |0.167 |

6) Based on meteorological records, the probability that it will snow in a certain town on

January 1st is 0.477. Find the probability that in a given year it will not snow on

January 1st in that town.

7) Determine whether the following is a probability distribution. If not, identify the

requirement that is not satisfied. A police department reports that the probabilities

that 0, 1, 2, 3, and 4 car thefts will be reported in a given day are 0.165, 0.298, 0.268,

0.161, and 0.072 respectively.

8) Of the 51 people who answered "yes" to a question, 14 were male. Of the 98 people

who answered "no" to the question, 6 people were male. Answer the following.

a) Write a contingency table for the data.

b) If one person is selected at random from the group, what is the probability that

the person answered "yes" or was male?

c) What is the probability that a male answered yes?

d) What is the probability that a person was a women and answered no?

9) The probability that a person has immunity to a particular disease is 0.2. Find the mean

number who have immunity in samples of size 17. At what number would you

consider the number people having the disease unusual, and why?

10) Find the probability that 4 randomly selected people all have the same birthday.

Ignore leap years.

11) Use the table of binomial probabilities to find the probability of at least 10 successes if

n = 11 and p = 0.6.

12) The manager of a bank recorded the amount of time each customer spent waiting in

line during peak business hours one Monday. The frequency table below summarizes

the results.

|Waiting Time (minutes) |Number of Customers |

|0-3 |11 |

|4-7 |14 |

|8-11 |9 |

|12-15 |4 |

|16-19 |8 |

|20-23 |3 |

|24-27 |2 |

a) If we randomly select one of the times represented in the table, what is the probability

that it is at least 12 minutes or between 8 and 15 minutes?

b) What is the probability that it is at most 19 minutes?

13) The number of lightning strikes in a year at the top of a particular mountain has a

Poisson distribution with a mean of 2.9. Find the probability that in a randomly

selected year, the number of lightning strikes is 3.

14) What important question must you answer before computing an "and" probability?

How does the answer influence your computation?

15) On an exam on probability concepts, Sue had an answer of -( . Explain how she

knew that this result was incorrect.

16) Do probability distributions measure what did happen or what will probably happen?

How do we use probability distributions to make decisions?

17) A game is said to be "fair" if the expected value for winnings is 0, that is, in the long

run, the player can expect to win 0. Consider the following game. The game costs $1

to play and the winnings (payoff minus your bet) are $5 for red, $3 for blue, $2 for

yellow, and nothing for white. The following probabilities apply. What are your

expected winnings? Does the game favor the player or the owner?

|Outcome |Probability |

|Red |0.02 |

|Blue |0.04 |

|Yellow |0.16 |

|White |0.78 |

18) a) Describe an event whose probability of occurring is 1 and explain what that

probability means.

b) Describe an event whose probability of occurring is 0 and explain what that

probability means.

19) What important question must you answer before computing an "or" probability?

How does the answer influence your computation?

20) A batch consists of 12 defective coils and 88 good ones. Find the probability of

getting two good coils when two coils are randomly selected if the first selection is

replaced before the second is made.

21) A company purchases shipments of machine components and uses this acceptance

sampling plan: Randomly select and test 25 components and accept the whole batch if

there are fewer than 3 defectives. If a particular shipment of thousands of components

actually has a 4% rate of defects, what is the probability that this whole shipment will

be accepted?

22) In a game you roll a fair die and you get $3.00 for a 1 or a 5 and nothing otherwise.

a) Write out the probability distribution.

b) What would you expect to pay to play this game?

c) How much would you expect your earnings to vary from the expected? (Hint:

Find the standard deviation.)

23) The number of calls received by a car towing service averages 9.6 per day (per 24-

hour period). After finding the mean number of calls per hour, find the probability that

in a randomly selected hour the number of calls is 3.

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