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uestion 3

A) Betty must pass the English, Mathematics and Science test, in order to be awarded the STEM Scholarship to
study in Australia. The chances of her passing English is 55%, failing Mathematics is 70% and failing Science is
40%.
a) Draw a probability tree to show all the possible outcomes for the three tests.
Use the probability tree above to answer the following questions.
Find the probability that she will:
b) pass two tests and fail one.
c) qualify for the scholarship.
d) pass at least one test.
e) fail at most two tests.
f) Only pass the science test

1 answer

To solve this problem, we first need to clarify the probabilities related to Betty's tests. Given the following probabilities:

  • Probability of passing English (P(E)) = 55% = 0.55
  • Probability of failing Mathematics (P(M')) = 70% = 0.70 → Probability of passing Mathematics (P(M)) = 1 - 0.70 = 0.30
  • Probability of failing Science (P(S')) = 40% = 0.40 → Probability of passing Science (P(S)) = 1 - 0.40 = 0.60

A) Probability Tree

The probability tree will have three levels corresponding to each of the tests: English, Mathematics, and Science.

  1. First Level (English):

    • Pass (E): Probability = 0.55
    • Fail (E'): Probability = 0.45

  2. Second Level (Mathematics):

    • From Pass (E):
      • Pass (M): Probability = 0.30
      • Fail (M'): Probability = 0.70
    • From Fail (E'):
      • Pass (M): Probability = 0.30
      • Fail (M'): Probability = 0.70

  3. Third Level (Science):

    • From Pass (EM):
      • Pass (S): Probability = 0.60
      • Fail (S'): Probability = 0.40
    • From Pass (EM'):
      • Pass (S): Probability = 0.60
      • Fail (S'): Probability = 0.40
    • From Fail (E'M):
      • Pass (S): Probability = 0.60
      • Fail (S'): Probability = 0.40
    • From Fail (E'M'):
      • Pass (S): Probability = 0.60
      • Fail (S'): Probability = 0.40

Outcomes of the Probability Tree:

Now we can label the outcomes based on the combinations of passing (P) and failing (F).

  1. P (E) P (M) P(S) - Pass English, Pass Mathematics, Pass Science (PPP)
  2. P (E) P (M) F(S) - Pass English, Pass Mathematics, Fail Science (PPF)
  3. P (E) F(M) P(S) - Pass English, Fail Mathematics, Pass Science (PFP)
  4. P (E) F(M) F(S) - Pass English, Fail Mathematics, Fail Science (PFF)
  5. F (E) P (M) P(S) - Fail English, Pass Mathematics, Pass Science (FPP)
  6. F (E) P (M) F(S) - Fail English, Pass Mathematics, Fail Science (F PF)
  7. F (E) F(M) P(S) - Fail English, Fail Mathematics, Pass Science (FFP)
  8. F (E) F(M) F(S) - Fail English, Fail Mathematics, Fail Science (FFF)

b) Probability of passing two tests and failing one

The outcomes where Betty passes 2 tests and fails 1 are: PPP (not possible), PPF, PFP, FPP.

  • PPF: (0.55)(0.30)(0.40) = 0.066
  • PFP: (0.55)(0.70)(0.60) = 0.231
  • FPP: (0.45)(0.30)(0.60) = 0.081

So, the total probability of passing 2 tests and failing 1 is: \[ P(PPF) + P(PFP) + P(FPP) = 0.066 + 0.231 + 0.081 = 0.378 \]

c) Probability that she qualifies for the scholarship

Betty qualifies for the scholarship if she passes all three tests (PPP).

  • PPP: (0.55)(0.30)(0.60) = 0.099

d) Probability of passing at least one test

To find the probability of passing at least one test, it's easier to calculate the probability of failing all tests and subtracting from 1.

  • P(Failure): \[ P(FFF) = (0.45)(0.70)(0.40) = 0.126 \] Thus, the probability of passing at least one test is: \[ P(\text{at least one pass}) = 1 - P(FFF) = 1 - 0.126 = 0.874 \]

e) Probability of failing at most two tests

Failing at most 2 tests means she can either pass all tests, pass 2 tests (which we previously calculated), or pass 1 test.

We already calculated:

  • Failing 3 tests = 0.126 (FFF)
  • Failing 2 tests = 0.378 (PPF + PFP + FPP)
  • Solutions for PPF, PFP, FPP in the previous parts are already taken into account.

Therefore, the total probability of failing at most 2 tests: \[ P(| \leq 2 failures) = 1 - P(0 failures) = 1 - 0.126 = 0.874. \]

f) Probability of passing only the science test

The only scenario Betty passes only the Science test is:

  • Fail English (F) and Fail Mathematics (F) but Pass Science (P): (F, F, P)

The calculation for this outcome: \[ P(FEP) = (0.45)(0.70)(0.60) = 0.189. \]

Summary of the Probabilities

  • b) Probability of passing two tests and failing one = 0.378
  • c) Probability of qualifying for the scholarship = 0.099
  • d) Probability of passing at least one test = 0.874
  • e) Probability of failing at most two tests = 0.874
  • f) Probability of only passing the science test = 0.189