01
Independence puzzle
Assume
, , and are mutually independent. Compute in terms of , , and .
Solution
07
(1) Use inclusion-exclusion principle.
(2) Use the fact that they are mutually independent.
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02
Pairwise independent, not mutually independent: three coin flips
Flip a coin three times in sequence. Label events like this:
- exactly one heads among first and second flips - exactly one heads among second and third flips - exactly one heads among first and third flips Verify that
are pairwise independent but not actually mutually independent.
Solution
08
(1) Find probabilities for individual events.
.
(2) Compute pairwise probabilities.
The only way
and happen is when the second flip is heads and the first and third flip are tails, and when the first and third are heads and when the second is heads. So, . The only way
and happen is when only the third flip is heads or when only the first and second flip are heads. So, . The only way
and happen is when only the first flip is heads or when only the second and third flip are heads. So, .
(3) Disprove mutual independence.
Link to original
, , and cannot happen simultaneously, so .
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.
Solution
04
Syntax errors vs. logic errors B
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.
Are the events “program has a syntax error” and “program has a logic error” independent? Justify your answer.
Solution