Question

at the beginning of the semester, a professor tells students that if they study for the tests, then there is a 55% chance they will get a b or higher on the tests. if they do not study, there is a 20% chance that they will get a b or higher on the tests. the professor knows from prior surveys that 60% of students study for the tests. the probabilities are displayed in the tree diagram. the professor informs the class that there will be a test next week. what is the probability that a randomly selected student studied if they do not pass the test with a b or higher? 0.45 0.46 0.54 0.59

Explanation:

Step1: Calculate the probability of not getting B or higher

• Probability of studying and not getting B or higher: \(0.6\times0.45 = 0.27\)
• Probability of not studying and not getting B or higher: \(0.4\times(1 - 0.20)=0.4\times0.80 = 0.32\)
• Total probability of not getting B or higher: \(P(\text{not B or higher})=0.27 + 0.32=0.59\)

Step2: Use Bayes' theorem

Bayes' theorem formula is \(P(A|B)=\frac{P(B|A)P(A)}{P(B)}\)
Let \(A\) be the event of studying and \(B\) be the event of not getting B or higher.
\(P(A)=\) probability of studying \(= 0.6\)
\(P(B|A)=\) probability of not getting B or higher given studying \(=0.45\)
\(P(B)=\) probability of not getting B or higher \(=0.59\)
\(P(A|B)=\frac{0.45\times0.6}{0.59}=\frac{0.27}{0.59}\approx0.46\)

Answer:

0.46