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Mathematics 1670
Assignment 2 – Chapters 5-8
Probability, Discrete, Binomial and Normal Probability Distribution
NameID SOLUTIONS__ Section: ___
Due Date: February 25, 2010
1. A manufacturing company has three factories: X, Y and Z. The daily output of each is shown here.
Product / Factory X / Factory Y / Factory Z / TotalTV’s / 18 / 32 / 15 / 65
Stereos / 6 / 20 / 13 / 39
Total / 24 / 52 / 28 / 104
a) Determine the marginal probabilities:
P (X) = __24/104P (Z) = __28/104P(Stereo)= _39/104
P (Y) = _52/104P(TV)= 65/104
b) Determine the joint probabilities:
P (Stereo and X) = 6/104
P (Yand TV) =32/104
P ( X and Z) =0
P ( XorZ) = 52/104
P( TV and Z) 15/104
P(Y and Stereo)20/104
c) Determine the conditional probabilities:
P ( TV | Y) = 32/52
P ( X| Stereo) =6/39
P ( Y| TV) =32/65
2. On New Year’s Eve the probability of a person driving while intoxicated is 0.32, the probability of a person having a driving accident is 0.09 and the probability of a person having a driving accident while intoxicated is 0.06.
a) What is the probability of a person driving while intoxicated or having a driving accident?
b) Draw a venn diagram to show the above probability.
a) P(I or A)=P(I) + P(A) –P(I and A) = 0.32 + 0.09 – 0.06 = 0.35
b)
3. Roll two dice and multiply the numbers together.
a) Write out the sample space.
1 / 2 / 3 / 4 / 5 / 61 / 1 / 2 / 3 / 4 / 5 / 6
2 / 2 / 4 / 6 / 8 / 10 / 12
3 / 3 / 6 / 9 / 12 / 15 / 18
4 / 4 / 8 / 12 / 16 / 20 / 24
5 / 5 / 10 / 15 / 20 / 25 / 30
6 / 6 / 12 / 18 / 24 / 30 / 36
b) What is the probability that the product is a multiple of 6? 15/36
c) What is the probability that the product is less than 9? 16/36
4At a local college, 54% of incoming students have computers. Construct a probability distribution for three students. Let X represent the number of students with a computer..
a) Find the Expected mean and Standard deviation.
b) What is the probability there is at least one student has a computer?
List 4
X0123
P(X)0.0970.3430.4020.157
a) Expected Value= np=3*0.54=1.62
Standard Deviation = =0.863
b) 1-0.097 = 0.903
5.Shown below is a discrete probability distribution. (It is not binomial)
x / P(x)0 / .2
1 / .3
2 / .25
3 / .1
4 / .15
Find:(Show how you arrived at your answers)
a.Probability x is at least 1. (Use the complement).1-0.2=0.8
b.P(x=4)=0.15
c.P(x=0)+P(x=1)+P(x=2)=0.75
dP( x is at most 2)P(x=0)+P(x=1)+P(x=2)=0.75
eThe expectation of the distribution.1.7_
fThe standard deviation of the distribution.1.31
6.A binomial probability distribution has n = 6and π = 0.35.
a) Show the complete probability distribution. You can use your calculator or the binomial formula.
b) Find the mean and standard deviation:µ = 2.1
σ =1.17
7.A binomial probability distribution has n = 12 and π = 0.6. Find the following. Using the your calculator
aP(x10)______
bP(x<7)______
cP( x is at least 3)______
dP( x is at most 9 )______
eP(x is between 7 and 12 inclusive)______
9. One thousand tickets are sold at $1, each for 4 prizes of $100, $50,$25, and $10. After each prize drawing the winning ticket is then returned to the pool of tickets. What is the expected value if a person purchases two tickets?Construct a probability distribution table for ALL events.
10.A factory produces computer chips with a 0.9% defect rate. In a batch of 100 computer chips, what is the probability that
a) only 1 is defective?
b) at least 3 are defective?
11.During October, the average temperature of Whitman Lake is 53.2o and the standard deviation is 2.3o. Assume the variable is normally distributed. For a randomly selected day in October, find the probability that the temperature will be
a) Between 49 and 55o
b) If the lake temperature were above 60o, would you call it very warm? Explain!
There is a 0.1% chance of the temperature being above 60
12.A cruise director schedules 4 different movies, 2 bridge games and 3 tennis games for a 2-day period. If a couple selects 3 activities, find the probability that they attend 2 movies and I tennis game.
One in four choice for the first movie then one in three choice unless you want to watch the movie again. Then a one in 2 choice of tennis. ¼ * 1/3 * ½ = 1/24
13.In a club there are 7 women and 5 men. A committee of 3 women and 2 men is to be chosen. How many different possibilities are there? ( Hint. The choosing of the women and the choosing of the men are different events).
14. For a certain year, the average annual salary in Pennsylvania was $24393. Assume that the salaries were normally distributed for a certain group of wage earners, and the standard deviation of this group is $4362.
- Find the probability that a randomly selected individual earned less than $26000
64%
- Find the probability that for a randomly selected sample of 25 individuals, the mean salary was less than $26000. 98%
15. The daily milk production of Guernsey cows is approximately normally distributed with a mean of 35 kg/day and a standard deviation of 6 kg/day.
a) What is the probability that a days production for a single animal will be less than 28 kg
b) The producer is concerned when the milkproduction of a cow falls below the 5th percentile because the animalmay be ill. How much milk will a cow in the 5th percentile produce (in kg) daily.
16. Which of the following exam scores has the better relative position? Explain.
aA score of 42 on a test with a mean of 39 and a standard deviation of 4.
bA score of 76 on a test with a mean of 71 and a standard deviation of 3.
Since b) is in the 95 percentile it is the better relative position.
Use the following to answer questions 17-20: For each question sketch the curve used. Show its mean and standard deviation.
Truck tire life is normally distributed with a mean of 60,000 miles and a standard deviation of 4,000 miles.
17.What is the probability that a tire will last 72,000 miles or more?
18.What is the probability that a tire lasts between 54,000 and 66,000 miles?
19.You bought four tires. What is the probability that the average mileage of the four tires exceeds 66,000 miles? Treat these tires as a sample.
20.For a set of four tires, what is the probability that the average tire life is between 57,000 and 63,000 miles? Treat these tires as a sample.