Question

there is a 60% that your math teacher will give a pop quiz tonight and due to the fact that you have a soccer game there is a 85% chance you will not be able to do your math homework.
you decide that you will not do the homework if the probability of there not being a pop quiz and you do your homework is greater than 5%.
what do you do and why?
a do your homework as the probability is 0.51
b do your homework as the probability is 0.06
c do not do your homework as the probability is 0.51
d do not do your homework as the probability is 0.06

Explanation:

Step1: Calculate the probability of no pop - quiz

The probability of a pop - quiz \(P(Q)=0.60\). So the probability of no pop - quiz \(P(
eg Q)=1 - P(Q)=1 - 0.60 = 0.40\)

Step2: Calculate the probability of doing homework

Case 1: Pop - quiz and do homework. The probability of pop - quiz and do homework \(P(Q\cap H)=P(Q)\times P(H|Q)\). But we are not given \(P(H|Q)\). However, we can use the formula for conditional probability in another way.
The probability of doing homework if there is no pop - quiz and we consider the decision rule (do homework if the probability of no pop - quiz and doing homework is greater than 5%).
Let's assume two cases:
If we do homework:
The probability of no pop - quiz and doing homework \(P_1 = P(
eg Q)\times1=0.40\) (assuming if we decide to do homework, the probability of doing homework is 1 in the case of no pop - quiz for the sake of calculation based on the decision rule)
If we do not do homework:
The probability of no pop - quiz and not doing homework \(P_2=P(
eg Q)\times0 = 0\) (assuming if we decide not to do homework, the probability of doing homework is 0)

Since \(0.40>0.05\)

Answer:

A. do your homework as the probability is \(0.40>0.05\) (where \(0.40\) is the probability of no pop - quiz (which is a condition related to the decision, and if we assume doing homework when we consider the non - pop - quiz situation as per the problem's decision logic of comparing probabilities greater than \(5\%\))