Due date: Tuesday 9/2, 11:59pm
Events and outcomes
01
03
Events - descriptions to sets
You are modeling quality assurance for cars coming off an assembly line. They are either good (G) or broken (B). You watch 4 cars come off and record their status as a sequence of these letters, for example ‘GGBG’.
Determine the sets defined by the events having the following descriptions:
(a) “third car is broken”
(b) “all cars have the same status”
(c) “at least one car is broken”
(d) “no consecutive cars have the same status”
Link to originalSolution
03
(a) Since only the third car is broken, and the other three cars can have any status, the relevant set
is (b) In this case, either all cars are good or all cars are broken. Therefore, the relevant set
is (c) In this scenario, the only combination not in the relevant set is ‘GGGG’. Therefore,
(d) In this scenario, given two cars
and , . Therefore, Link to original
Probability models
02
06
Researcher’s degree
Of 1000 researchers at a research laboratory, 375 have a degree in mathematics, 450 have a degree in computer science, and 150 of the researchers have a degree in both fields. One researcher’s name is selected at random.
(a) What is the probability that the researcher has a degree in mathematics, but not in computer science?
(b) What is the probability that the researcher has no degree in either mathematics or computer science?
Link to originalSolution
06
(a)
(b)
Link to original
03
02
Inclusion-exclusion reasoning
Your friend says: “according to my calculations, the probability of
is and the probability of is , but the probability of and both happening is only .” You tell your friend they don’t understand probability. Why?
Link to originalSolution
05
(1) State the inclusion-exclusion principle.
(2) Examine possibilities based on given values.
Given that
is and is , we have that . Since
, we know that . Therefore,
Link to original.
Conditional probability
04
02
Conditioning - two dice, at least one is 5
Two dice are rolled, and at least one is a 5.
What is the probability that their sum is 10?
Link to originalSolution
09
(1) Let
be the outcome of the first die and be the outcome of the second die. We are asked to compute
(2) Compute individual probabilities.
There is only one combination out of the 36 possible combinations of two dice rolls in which at least 1 die rolls a 5 and both sum up to 10 (5, 5).
There are
combinations of dice rolls in which at least one is a .
(3) Plug into formula.
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05
03
Conditioning - two dice, differing numbers
Two dice are rolled, and the outcomes are different.
What is the probability of getting at least one 1?
Link to originalSolution
10
(1) Let
be the outcome of the first die and be the outcome of the second die. We are asked to compute
(2) Compute individual probabilities.
There are
combinations in which at least one die rolled a . Since one of these combinations is , we have 10 combinations in which the outcomes are unequal. There are
combinations in which the outcome of the two dice differ.
(3) Plug into formula.
Link to original
06
06
Conditioning relation
Suppose you know
and and . Calculate
and and . Link to originalSolution
15
(1) Set up conditional probability formula.
Solve for
.
(2) Plug in given values.
(3) Set up conditional probability formula.
Solve for
.
(4) Plug in given values.
(5) Use inclusion-exclusion principle to find
. Link to original