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Let's break down each part of the problem step by step.

### Part A: Probability Tree for Betty's Tests

#### Key information:
- The chances of passing English = 55% (or 0.55).
- The chances of failing Mathematics = 70% (or 0.70), so passing Mathematics = 30% (or 0.30).
- The chances of failing Science = 40% (or 0.40), so passing Science = 60% (or 0.60).

We need to create a probability tree that represents these probabilities for passing or failing each test. The tree will have three branches, one for each test, with their corresponding probabilities.

#### Steps:
1. The first branch is for passing or failing English.
2. Each outcome of English has a sub-branch for passing or failing Mathematics.
3. Each outcome of the Math branch has a sub-branch for passing or failing Science.

### Part B: Questions Based on the Tree

From the probability tree, we can calculate the probabilities for the following outcomes:

1. **Pass two tests and fail one**  
   This occurs in any of the following combinations:
   - Pass English, pass Math, fail Science.
   - Pass English, fail Math, pass Science.
   - Fail English, pass Math, pass Science.

2. **Qualify for the scholarship**  
   To qualify, she must pass all three tests. So, we calculate the probability of passing all tests.

3. **Pass at least one test**  
   We calculate the complement of failing all tests, which means subtracting the probability of failing all three from 1.

4. **Fail at most two tests**  
   This includes passing one or more tests, so we calculate the sum of probabilities for passing one, two, or three tests.

5. **Only pass the science test**  
   This is the scenario where she fails both English and Math but passes Science.

### Part B: Probability of Winning or Losing the Race

#### Key information:
- Probability of losing the first race = \( \frac{2}{5} \).
- Probability of winning the second race is twice the probability of losing it.

Let the probability of losing the second race be \( p \). Thus, the probability of winning the second race = \( 2p \). 

We are asked to find the probability of winning one race and losing the other.

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Let me start with the probability tree diagram for part A and then calculate the probabilities for each part of the question.

Betty must pass the English, Mathematics, and Science tests in order to be awarded the STEM Scholarship. The chances of passing English is 55%, failing Mathematics is 70%, and failing Science is 40%. Based on these probabilities, a probability tree needs to be drawn to show all possible outcomes. Additional questions include finding the probabilities of passing or failing combinations of tests, and determining probabilities related to a separate racing event for Toea, involving two races with certain win/loss probabilities.

Solve a probability problem involving STEM scholarship qualification based on passing English, Mathematics, and Science tests. Learn how to draw probability trees and calculate outcomes for various combinations of passed and failed tests. Also, find the probability of winning one race and losing the other based on given conditions. Suitable for Grades 10-12 students.

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