Question

1- You take a test, and 60% pass the first time and 40% fail.
The second time around,80% pass the test and 20% fail.
The third time, 90% pass and 10% fail.

a. What is the probability that you pass on the 2nd time?
b. What is the probability that you pass on the 3rd time?
c. What is the probability you fail?

Answer 1

Do you keep taking the tests, whether you pass or fail prior test(s)?

If only those who fail take re-tests, which seems likely, then this is a series of statistically independent events.

Passing the second time means failing the first (0.40) and passing the second (0.80), or a joint probability of (0.40)(0.80)= 0.32.

Etc.

Answer 2

When you have a series of independent events, such as a series of tests, then the probabilities for their joint outcomes are multiplicative.

Tossing three heads with a coin = (1/2)(1/2)(1/2)

Getting a 4 with a die the first time but not the second = (1/6)(5/6)