01
Conditioning - restrict to 4th-year students
Student test-passing rates, by year:
1st year 2nd year 3rd year 4th year Pass 0.155 0.340 0.255 0.160 Fail 0.025 0.040 0.015 0.010 What is the likelihood that a randomly chosen 4th-year student passed the test? What about for 1st-year students?
Solution
08
(1) We are asked to compute
. Set up the conditional probability formula.
(2) We have from the table that
and . Therefore,
(3) For 1st-year students, we have
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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?
Solution
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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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?
Solution
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.
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04
Multiplication - drawing two hearts
Two cards are drawn from a standard deck (without replacement).
(a) What is the probability that both are hearts?
(b) What is the probability that both are 4?
Solution
16
(a)
Let
be the outcome of the first card, be the outcome of the second card, and denote “hearts”. Since there are 52 cards in a standard deck with 13 of them being hearts, we have (b)
Similarly to part (a), we have
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05
Conditional probability - algebra games
Assume that
, , and partition the sample space, and assume this data:
Find these values:
Solution
01
(1) Use the Law of Total Probability to find
.
(2) Use Bayes’ Theorem to find
.
(3) Use Bayes’ Theorem to find
.
(4) Use Bayes’ Theorem to find
. Link to original
06
Conditioning relation
Suppose you know
and and . Calculate
and and .
Solution
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
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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?
Solution
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
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?
Solution
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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09
Syntax errors vs. logic errors A
A computer program may contain a syntax error or a logic error or both types of errors. The probability that a program has both types of error is 0.16. The probability that a program has a syntax error given that it has a logic error is 0.4. The probability that a program has a logic error given that it has a syntax error is 0.5.
Find the probability that a particular program has at least one type of error.
Solution
03
Let A be a syntax error and B a logic error.
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