Problems

01

Independence algebra

Assume $A_1$, $A_2$, and $A_3$ are mutually independent. Compute $P[A_1{A}_2\cup A_3]$ in terms of $P[A_1]$, $P[A_2]$, and $P[A_3]$.

02

Pairwise independent, not mutually independent: three coin flips

Flip a coin three times in sequence. Label events like this:

• $A$ – exactly one heads among first and second flips
• $B$ – exactly one heads among second and third flips
• $C$ – exactly one heads among first and third flips

Verify that $A,B,C$ are pairwise independent but not actually mutually independent.

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.

04

Syntax errors vs. logic errors, Part 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.