  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

To create a probability tree for Betty's chances of passing the three tests (English, Mathematics, and Science), we will first need to clarify the probabilities and the events associated with them:

- The probability of passing English is 55%, so \( P(\text{Pass English}) = 0.55 \) and \( P(\text{Fail English}) = 0.45 \).
- The probability of failing Mathematics is 70%, so \( P(\text{Fail Math}) = 0.70 \). Therefore, \( P(\text{Pass Math}) = 0.30 \).
- The probability of failing Science is 40%, so \( P(\text{Fail Science}) = 0.40 \). Therefore, \( P(\text{Pass Science}) = 0.60 \).

We will now outline the structure of the probability tree, which will have branches for each test, starting with English, followed by Mathematics, and finally Science.

### Probability Tree Structure:

1. **First branch: English**
   - Two outcomes: Pass English (0.55) or Fail English (0.45)

2. **Second branch: Mathematics** (for each English outcome)
   - From Pass English: Pass Math (0.30) or Fail Math (0.70)
   - From Fail English: Pass Math (0.30) or Fail Math (0.70)

3. **Third branch: Science** (for each Math outcome)
   - From each combination of English and Math outcomes, two possible Science outcomes:
     - Pass Science (0.60) or Fail Science (0.40)

This results in 8 total branches representing all possible combinations of outcomes for the three tests.

I can create a visual representation if you upload a document or image for further guidance. Would you like me to explain how to calculate the probabilities of each final outcome or help in drawing the tree in more detail?

### Related Questions:
1. How would you calculate the probability of passing all three tests?
2. What is the probability that Betty fails at least one test?
3. If Betty passes English and Math, what is the probability that she also passes Science?
4. What is the probability that Betty passes exactly two out of the three tests?
5. How do the results change if the probability of passing Math increases to 50%?

### Tip:
When constructing probability trees, it's useful to multiply the probabilities along each branch to find the joint probability of each outcome.

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%. Draw a probability tree to show all the possible outcomes for the three tests.

Explore how to draw a probability tree for Betty's chances of passing English, Mathematics, and Science tests with given probabilities. This problem covers core probability concepts such as conditional probability and the multiplication rule, suitable for students in Grades 9-11.

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