Question

Alex has 3 tests to take this week. The probability that Alex passes the math test is 0.63 , the probability Alex passes the History test is 0.80, and the probability Alex passes the chemistry test is 0.88

1. (4 pts) create a complete tree diagram for the three tests- let T be passed tests
2. (4 pts) create a probability distribution for the number of tests passed
3. what is the probability that Alex will fail all three tests
4. what is the probability that Alex will pass at least 2 tests

Answer 1

a) The tree diagrams here are given as:

b) The PDF for number of tests passed is computed here as:

P(T = 3) = P(M pass)P(H pass) P( C pass) = 0.63*0.8*0.88 = 0.44352

P(T = 0) = P(M fail)P(H fail)P(C fail) = 0.37*0.2*0.12 = 0.00888

P(T = 2) = P(MH pass)P(C fail) + P(MC pass)P(H fail) + P(HC pass)P(M fail)
= 0.63*0.8*0.12 + 0.63*0.88*0.2 + 0.8*0.88*0.37
= 0.43184

Therefore P(T = 1)
= 1 - P(T = 0) - P(T = 2) - P(T = 3)
= 1 - 0.43184 - 0.00888 - 0.44352
= 0.11576

Therefore the PDF for T here is given as:
P(T = 0) = 0.00888
P(T = 1) = 0.11576
P(T = 2) = 0.43184
P(T = 3) = 0.44352

c) From the above part, we have:
P(T = 0) = 0.00888

Therefore 0.00888 is the probability of him failing all the 3 tests.

d) P(T >= 2) = P(T = 2) + P(T = 3) = 0.43184 + 0.44352 = 0.87536

Therefore 0.87536 is the required probability here.