Due date: Tuesday 9/16, 11:59pm
Repeated trials
01
04
Guessing on a test
Your odds of getting any given exam question right are
. The exam has 4 questions, and you need to answer 3 correctly to pass. (a) What is the probability that you pass?
(b) After finishing the exam, you are 100% sure that you got the second question right. Now what are the odds that you pass?
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08
(1) The probability that you pass follows a binomial distribution.
since there are 4 questions to account for. We choose
questions to be correct, where or .
(2) Describe the situation wherein you get the second question right.
Now, out of the remaining three questions, you need to answer at least two correctly.
This too follows a binomial distribution.
(3) Set up binomial distribution formula.
, since we only need to account for the remaining 3 questions. We choose
questions to be correct, where or . Link to original
02
01
Independent trials - At least 45 good paper clips
For a paper clip production line, 90% of the paper clips come off good, and 10% come off broken.
You buy a box of 50 paper clips from this line. What is the probability that at least 45 of them are good?
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05
(1) State the formula for a binomial distribution.
(2) State parameters of binomial distribution.
since you buy paper clips.
ranges from to .
.
.
(3) Use summation notation to find
. Link to original
Reliability
03
01
Reliability for complex process
Consider a process with the following diagram of components in series and parallel:
Use
to denote the event that component succeeds. Suppose the success probabilities per component are given by this chart:
1 2 3 4 5 6 7 8 80% 60% 40% 90% 80% 50% 70% 90% What are the odds of success for the whole process?
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09
(1) Describe dependencies of components.
The series 4, 5 runs parallel to component 3.
The components 2, 3, 4, 5, 6 all run in a series.
The latter is parallel to component 7
The circuit starts at component 1 and ends at component 8
(2) Find the probability the component 4 and 5 succeeded. (Denote as
)
(3) Find the probability 3, and
succeed. Note that these are in parallel. (Denote as )
(4) Find the probability 2,
, and 6 succeed. (Denote as )
(5) Find the probability 7 and
succeed. Note that these are in parallel. (Denote as )
(6) Lastly, find the probability
, , and 8 succeed. (Denote as ). Link to original
Discrete random variables
04
03
PMF from CDF for tracking gasoline
I am trying to keep track of how much gasoline (rounded to the nearest integer and denoted by
) my car uses every week. I have managed to find the CDF of : Assume gasoline costs $3 a gallon. Let
denote the amount of money I spend on gasoline every week. What is the PMF of ? Link to originalSolution
18
First find the PMF of
:
0 1 2 3 0.1 0.1 0.3 0.5 Now the PMF of
: Link to original
0 3 6 9 0.1 0.1 0.3 0.5
05
01
Digit of a real number
Suppose a real number is chosen randomly in the unit interval
. Consider the decimal expansion of this number. Let be a random variable giving the first digit after the decimal point. Find the possible values, the PMF, and the CDF of . Link to originalSolution
10
(1) Find the possible values of
. After the decimal point, any of the 10 digits can appear, so
.
(2) Find the PMF of
. Each digit has a
chance of appearing. So, the PMF of is
(3) Find the CDF of
. The CDF of
is given as follows.
(4)
Listing out the individual cumulative probabilities for each
Link to originalis also acceptable.
Review problems
06
06
Multinomial - Colored marbles in a line
How many ways are there to line up 10 colored marbles (2 red, 3 white, 5 blue), assuming you cannot distinguish marbles of the same color?
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03
Use the multinomial coefficient.
Use that
. We have three bins,
, , and . Thus,
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07
07
Multiplication Rule - Fund performance
The odds of the Winning Fund outperforming the market in a random year are 15%. The odds that it outperforms the market in a 1-year period assuming it has done so in the prior year are 30%.
What is the probability of the Winning Fund outperforming the market in 2 consecutive years?
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02
(1) Define events.
Let
be the event where the Winning Fund outperforms the first year. Let
be the event where the Winning Fund outperforms the second year. We are asked to compute
.
(2) Find relevant probabilities.
.
.
(3) Compute
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08
08
Applicant qualifications A
A hiring manager will randomly select two people from a group of 5 applicants. Of the 5 applicants, 2 are more qualified and 3 are less qualified (but the manager does not know this).
If at least one of the less qualified applicants is selected, what is the probability that both applicants selected will be less qualified?
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02
Label the two ‘more qualified’ as A, B, and the three ‘less qualified’ as C, D, E.
9 options have at least one less qualified. 3 options have both less qualified.
Answer
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03
Applicant qualifications B
A hiring manager will randomly select two people from a group of 5 applicants. Of the 5 applicants, 2 are more qualified and 3 are less qualified (but the manager does not know this).
Let Event A be selecting 2 more qualified applicants and Event B be selecting 2 less qualified applicants. Determine whether A and B are independent events and justify your answer.
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