3

Question

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.
B) On a particular day, Toea had to run two races. The probability of losing the first race is 
5
2
. 
Find the probability of winning one race and losing the other race if the probability of winning the second race is 
twice the probability of losing the second race.

GPT-4o mini · Answer

Let's break this problem into two parts based on your description.

Part A: Betty's Test Probability Tree

a) Probability Tree

Let's first define the probabilities:

Probability of passing English (P(E)) = 0.55
Probability of failing English (F(E)) = 0.45
Probability of passing Mathematics (P(M)) = 0.30 (1 - 0.70)
Probability of failing Mathematics (F(M)) = 0.70
Probability of passing Science (P(S)) = 0.60 (1 - 0.40)
Probability of failing Science (F(S)) = 0.40

The probability tree will have levels for each test:

              Starts
            /       \
        English      F(E)
         /  \       /  \
     P(E)   F(E)  P(E)  F(E)
       |         |       |
       |       Math     Math  
      P(M)    F(M)    P(M)  F(M)
     /    \        |       | 
   P(S)   F(S)     P(S)  F(S)
   /  \   /  \     /  \   /  \
pass pass fail fail fail pass pass fail

Each branch can be multiplied to find the probabilities.

b) Pass Two Tests and Fail One

To find the probability that she will pass two tests and fail one, we need to consider the combinations:

Pass English, Pass Mathematics, Fail Science (P(E) * P(M) * F(S))
Pass English, Fail Mathematics, Pass Science (P(E) * F(M) * P(S))
Fail English, Pass Mathematics, Pass Science (F(E) * P(M) * P(S))

Calculating these:

\( P(E) * P(M) * F(S) = 0.55 * 0.30 * 0.40 = 0.066 \)
\( P(E) * F(M) * P(S) = 0.55 * 0.70 * 0.60 = 0.231 \)
\( F(E) * P(M) * P(S) = 0.45 * 0.30 * 0.60 = 0.081 \)

Now add the probabilities together: \[ P(\text{pass 2, fail 1}) = 0.066 + 0.231 + 0.081 = 0.378 \]

c) Qualify for the Scholarship

She qualifies for the scholarship if she passes all three tests. This probability is: \[ P(E) * P(M) * P(S) = 0.55 * 0.30 * 0.60 = 0.099 \]

d) Pass at Least One Test

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

The probability of failing all three: \[ F(E) * F(M) * F(S) = 0.45 * 0.70 * 0.40 = 0.126 \]

Thus, the probability of passing at least one test is: \[ P(\text{at least one}) = 1 - P(\text{fail all}) = 1 - 0.126 = 0.874 \]

e) Fail at Most Two Tests

Failing at most two tests includes:

Pass all three tests
Fail one test
Fail two tests

We already calculated P(passing all three) = 0.099.

The probabilities for failing one test were calculated in part b = 0.378.

Now, we calculate the probabilities for failing two tests:

Fail English, Fail Mathematics, Pass Science (F(E) * F(M) * P(S)) = 0.45 * 0.70 * 0.60 = 0.189
Fail English, Pass Mathematics, Fail Science (F(E) * P(M) * F(S)) = 0.45 * 0.30 * 0.40 = 0.054
Pass English, Fail Mathematics, Fail Science (P(E) * F(M) * F(S)) = 0.55 * 0.70 * 0.40 = 0.154

Adding them up: \[ P(\text{fail2}) = 0.189 + 0.054 + 0.154 = 0.397 \]

Now, total probability of failing at most two tests: \[ P(\text{at most 2}) = P(\text{all pass}) + P(\text{fail 1}) + P(\text{fail 2}) = 0.099 + 0.378 + 0.397 = 0.874 \]

f) Only Pass the Science Test

The probability of passing only Science (failing English and Mathematics) is: \[ F(E) * F(M) * P(S) = 0.45 * 0.70 * 0.60 = 0.189 \]

Part B: Toea's Race Probability

Let \( P(L1) = \frac{5}{2} \) for the first race, meaning the probability of winning \( P(W1) = 1 - P(L1) = 1 - \frac{5}{2} = -1.5 \). This is incorrect because a probability can't exceed 1.

However, if the scenario states that the probability of losing the first race is 0.2, then we can think of \( P(W1) = 0.8 \).

Let \( P(L2) \) be the losing probability of the second race, then we know \( P(L2) + P(W2) = 1 \) and \( P(W2) = 2 * P(L2) \).

Let’s denote \( P(L2) = x \) then \( P(W2) = 2x \).

Setting up the equation: \[ x + 2x = 1 \to 3x = 1 \to x = \frac{1}{3} \]

Thus:

\( P(L2) = \frac{1}{3} \)
\( P(W2) = 2 * \frac{1}{3} = \frac{2}{3} \)

Now, we want to find the probability of winning one race and losing the other. This could happen in two different ways:

Win first race, lose second: \( P(W1) * P(L2) = 0.8 * \frac{1}{3} = \frac{8}{30} = \frac{4}{15} \)
Lose first race, win second: \( P(L1) * P(W2) = 0.2 * \frac{2}{3} = \frac{4}{30} = \frac{2}{15} \)

Adding these scenarios together: \[ P(\text{winning one and losing the other}) = \frac{4}{15} + \frac{2}{15} = \frac{6}{15} = \frac{2}{5} \]

Summary of Answers:

b) Probability of pass 2 tests and fail 1: 0.378
c) Qualify for the scholarship: 0.099
d) Pass at least one test: 0.874
e) Fail at most two tests: 0.874
f) Only pass the science test: 0.189
Probability of winning one race and losing the other: \(\frac{2}{5}\)