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?
Solution
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
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02
Geometric wait time - Takes 10 rolls to get 6
A fair die is rolled until a six comes up. What are the odds that it takes at least 10 rolls?
Hint: you might find it easier to compute the odds of the complementary event.
Solution
06
(1) Describe the situation.
Let
. If it takes at least
rolls for a to come up, then the first rolls resulting in a number that is not a six.
(2) Compute relevant probabilities.
The probability that you roll a six is
. The probability that you do not roll a six is
.
(3) Set up formula.
(4) Use the formula for geometric series to evaluate the sum.
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03
Intersection accidents
Suppose that the odds of an accident occurring on any given day at the intersection of Ivy and Emmet is 0.05.
(a) What are the odds of a perfect week? (No accidents.)
(b) What are the odds of exactly 2 accidents in 30 days?
(c) What are the odds of the first accident occurring after day 4 and by day 10?
Solution
07
(a)
The odds of a perfect week follows a binomial distribution.
since there are days to account for. We choose
of them to have accidents.
(b)
(1) The odds of exactly 2 accidents in 30 days follows a binomial distribution.
since there are 30 days to account for. We choose
of these days to have accidents.
(2)
(c)
(1) Describe the situation.
We want four perfect days before an accident occurs.
Over the course of the next six days, we want at least one accident occurring.
Since these are independent trials, multiplying the individual probabilities together will yield the desired value.
(2) Set up the first condition.
since we want to account for 4 days. We choose
of these days to have accidents.
(3) Set up second condition.
since we now want to account for the next six days. We choose
of these days to have accidents, where ranges from through .
(4) Multiply the two expressions.
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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?
Solution
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
05
Winning the lottery
Suppose a lottery game requires that you purchase a $10 game card and advertises a 10% probability of winning a prize.
If you purchase 20 of these game cards, what is the probability you will win at least once?
Solution
06
Watching the Superbowl
A representative from Nielsen ratings randomly selects people in Charlottesville, VA and asks them whether they watched the Superbowl. The probability that any individual in Charlottesville watched the Superbowl is 0.3.
(a) What is the probability that if the representative asks 10 people, that less than 2 of them will have watched the Superbowl?
(b) What is the probability that the representative will have to ask at least 3 people to find someone who watched the Superbowl?
Solution
13
(a)
(b)
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