Ch 2, Probability and Bayes’ Rule

Please read the additional materials and answer the questions. Thank you Following don’t read. (Please see the attached writeup. It should not be too long to do. However, if you have questions regarding the translation of English to probability notation, please put it up on discussion so everyone can talk about the best way to make the translation. This is an important skill to have, especially when we get to chapter 4 use of Bayes’ rule to update prior probabilities. )
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BUS 190
HW 8 Probability
Passenger vehicle drivers in fatal crashes by age and crash type, 2015
Single vehicle
Multiple vehicle
Age
Number
Number
16-19
1,256
1,642
20-24
2,375
2,838
25-29
1,863
2,514
30-59
6,511
11,112
60-69
1,334
2,554
70+
1,193
2,700
1. Above are the frequencies of drivers of passenger vehicle drivers who were involved in fatal
accidents by age, for the year 2015.
LEAVE ALL YOUR WORK AS FRACTIONS SO I CAN SEE WHERE YOU ARE GETTING YOUR
NUMBERS FROM. DO NOT REDUCE!
a. Write down all the marginal totals and the grand total above. (10 pts) (Give one example
calculation for the rows and one for the columns)
For the following use S for single vehicle, M for multiple vehicle, and the age group appended to a
variable Age (i.e., Age16-19, Age20-24, etc.) for the age categories.
Write out the probability being asked for IN NOTATION using the notation above, then give the answer,
leaving the numerator and denominator in fractions. SHOW YOUR WORK. DON’T REDUCE THE
FRACTIONS.
b. What is the chance a driver involved in a fatal accident in 2015 is 16-19 years old? (2 pts.)
c. What is the chance a driver involved in a fatal accident was involved in a single vehicle accident?
(2 pts.)
d. What is the chance that a driver involved in a fatal accident was 16-19 years old AND involved n
a single vehicle accident? (show work, leave as fraction). (2 pts. )
e. What is the chance that a driver involved in a fatal accident was 16-19 years old OR involved in a
single vehicle accident? (show work, leave as fractions). (6 pts.)
f.
Given a driver was involved in a single vehicle fatal accident, what is the chance s/he was 16-19
years old?(HINT: CONDITIONAL) (2 pts.)
g. What is the chance that a driver aged 16-19 years old who was involved in a fatal accident was
involved in a single vehicle fatal accident? (HINT: CONDITIONAL) (2 pts.)
h. Are age and type of accident (single or multiple vehicle) independent? (You only have to
show the proper relationship holds for one cell – the rest fall into place.) State which
criterion you are using to show it.
2. Practicing identification and notation for unconditional and conditional probabilities
Write the following in proper notation for either unconditional or conditional probability.
You are a marketing researcher. Your population is the customers for your company. You are
interested in the following demographics (with the notation you are to use following)
Female = F
Male = M
Marital Status = S (for status)
Annual income level between $50K and $75: I_50_75
Age between 18 and 24: A_18_24
You will NOT have numbers for the probabilities below – just write the English statement in
probability notation, using the notation for events given above.
a. Probability that a randomly selected customer is between 18 and 24 years of age
b. Probability that a female customer being between 18 and 24 years of age
c. Probability that a customer who is between 18 and 24 years of age is female?
d. Probability that a customer makes between $50K and $75K inclusive, per year?
e.Probability, given a married male customer, what is the chance he makes between $50K
and$75K inclusive, per year?
f. Probability that a married female customer makes between $50K and $75K inclusive, per year
(you can use &, or, etc in describing someone)
3. You own and run a retail clothing store and want to see what is selling or not. Write out the
following in probability notation using the notation for the events given below. Put the given
numbers to your probabilities as appropriate. Your sales are split 50-50 between men’s and
women’s clothing.
Notation: W for women’s garments
M for Men’s garments
L for Large size garment
P for Pink garment
a.
25% of your sales are of large size garments.
b.
If a sale is of a large-size garment, the chance it is a men’s garment is 60%.
c. If a sale is of a woman’s garment, the chance it is large size is 30%.

d. . When a sale is of a pink garment, the chance it is a women’s garment is 90%.
e.
If a sale is of a women’s garment, the chance it is pink is 10%.
f.
The chance a garment sold is pink is 3%.
g. For the questions above – is your population customers or garments sold?
4. Bayes’ Rule
Dana is afraid that she will fail her QBA final, given her current grade. She believes that if she does not
find a tutor to help her, her chance of failing is 40%. However, if she does find a tutor, her chance of
failing drops to 10%. However, it is late in the semester, and she believes the chance she will be able to
find a tutor to help her is only 50%.
If Dana fails the final, what is the chance she had found a tutor?
Use the following notation:
F= fail the course
P = passes the course
T = finds a tutor
Tc = does not find a tutor
Using the above notation, set up a PROBABILITY tree using the information in the paragraph above. Be
sure to label ALL the branches with both the notation for the probability AND the numerical value of
that probability
Then multiply the probabilities across the branches. Give both the numerical value AND the probabilities
they represent in notation.
Finally, use the definition of conditional probability and the results from your tree calculations to
calculate the chance Dana had found a tutor if she fails the final.

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