The order in favor of Ann clearing a driving test is 1:4. The odds against Brad clearing the same driving test is 5:4. What is the probabilty that at least one of the them would clear the test ?

Answer

To solve the problem, we need to find the probability that at least one of Ann or Brad clears the driving test. We will break down the solution step by step.

### Step 1: Understand the Odds
The odds in favor of Ann clearing the test are given as 1:4. This means:
- For every 1 time she clears the test, there are 4 times she does not clear it.

For Brad, the odds against him clearing the test are given as 5:4. This means:
- For every 5 times he does not clear the test, there are 4 times he clears it.

### Step 2: Convert Odds to Probabilities
**For Ann:**
- The total parts = 1 (clearing) + 4 (not clearing) = 5 parts.
- Probability of Ann clearing the test (P(A)) = Number of favorable outcomes / Total outcomes = 1/5.
- Probability of Ann not clearing the test (P(A')) = 4/5.

**For Brad:**
- The total parts = 5 (not clearing) + 4 (clearing) = 9 parts.
- Probability of Brad clearing the test (P(B)) = 4/9.
- Probability of Brad not clearing the test (P(B')) = 5/9.

### Step 3: Calculate the Probability that None of Them Clears the Test
To find the probability that at least one of them clears the test, we first calculate the probability that neither of them clears the test (P(A') and P(B')).

- P(A' and B') = P(A') * P(B') = (4/5) * (5/9).

Calculating this:
\[ P(A' \text{ and } B') = \frac{4}{5} \times \frac{5}{9} = \frac{20}{45} = \frac{4}{9}. \]

### Step 4: Calculate the Probability that At Least One Clears the Test
Now, we can find the probability that at least one of them clears the test using the complement rule:
\[ P(\text{at least one clears}) = 1 - P(A' \text{ and } B'). \]

Substituting the value we found:
\[ P(\text{at least one clears}) = 1 - \frac{4}{9} = \frac{5}{9}. \]

### Final Answer
Thus, the probability that at least one of them clears the test is:
\[ \frac{5}{9}. \]