Due date: Tuesday 2/3, 11:59pm
Repeated trials
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Link to originalGuessing 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?
Solution
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Link to originalIndependent 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?
Solution
Reliability
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Link to originalReliability 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?
Solution
Discrete random variables
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Link to originalPMF 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 ?
Solution
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Link to originalGambling with a coin
Two players, A and B, are flipping a fair coin together. If it comes up heads, A pays $1 to B, and if it comes up tails, B pays $1 to A.
They play five rounds. Let
be a random variable recording A’s final winnings. (a) Find the set of possible values of
. (I.e., the set of outcomes with nonzero probability.) (b) Find the PMF and CDF of
.
Solution
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Link to originalDigit 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 .
Solution
Review problems
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Link to originalMultinomial - 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?
Solution
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Link to originalMultiplication 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?
Solution
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Link to originalApplicant 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?
Solution
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Link to originalApplicant 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.
Solution