a.) Rocco and Bliff are two koala bears participating in a series of animal behaviour tests. they each have 10 min to solve the maze. Rocco has an 85% probability of succeeding if he can smell the eucalyptus treat at the other end. He can smell the treat 60% of the time. Bliff has a 70% chance of smelling the treat, but when he does, he can solve the maze only 75% of them time. Neither bear will try to solve the maze unless he smells the eucalyptus. determine which koala bear is more likely to enjoy a tasty treat on any given trial. b.) explain how  you arrived at your conclusion.

To determine which koala bear, Rocco or Bliff, is more likely to succeed in solving the maze and enjoying a tasty treat, we can calculate the probability of each bear successfully solving the maze on a given trial.

### Part (a): Probabilities of Success for Rocco and Bliff

#### Rocco:
- Rocco can smell the eucalyptus treat 60% of the time, which is a probability of \( P(\text{Smell}) = 0.60 \).
- Given that he can smell the treat, Rocco has an 85% chance of solving the maze, which is a probability of \( P(\text{Solve} \mid \text{Smell}) = 0.85 \).

The probability that Rocco successfully smells the eucalyptus and solves the maze is the product of these two probabilities:
\[
P(\text{Success for Rocco}) = P(\text{Smell}) \times P(\text{Solve} \mid \text{Smell}) = 0.60 \times 0.85 = 0.51
\]
Thus, Rocco has a 51% chance of enjoying the treat on any given trial.

#### Bliff:
- Bliff can smell the eucalyptus treat 70% of the time, which is a probability of \( P(\text{Smell}) = 0.70 \).
- Given that he can smell the treat, Bliff has a 75% chance of solving the maze, which is a probability of \( P(\text{Solve} \mid \text{Smell}) = 0.75 \).

The probability that Bliff successfully smells the eucalyptus and solves the maze is the product of these two probabilities:
\[
P(\text{Success for Bliff}) = P(\text{Smell}) \times P(\text{Solve} \mid \text{Smell}) = 0.70 \times 0.75 = 0.525
\]
Thus, Bliff has a 52.5% chance of enjoying the treat on any given trial.

### Part (b): Conclusion

- Rocco has a 51% chance of successfully solving the maze and getting the treat.
- Bliff has a 52.5% chance of successfully solving the maze and getting the treat.

Since \( 0.525 > 0.51 \), **Bliff** is slightly more likely to enjoy a tasty treat on any given trial compared to Rocco.

### How I arrived at the conclusion:
1. I identified the relevant probabilities for both Rocco and Bliff:
   - The probability of each koala smelling the treat.
   - The conditional probability of solving the maze, given that the koala can smell the treat.
2. I multiplied these probabilities to find the overall likelihood of each koala succeeding.
3. I compared the results to see which koala had the higher chance of success.

Would you like further details or have any questions?

Here are some related questions for further exploration:
1. How would the probabilities change if Rocco's probability of smelling the treat increased to 70%?
2. What is the combined probability that neither Rocco nor Bliff solves the maze in one trial?
3. How does the difference in success rates between the two koalas change if their maze-solving skills were equal?
4. If each koala had two chances to solve the maze, what would be their respective probabilities of success?
5. How would Rocco’s success probability be affected if he had a better chance of solving the maze without smelling the treat?

**Tip:** In probability, multiplying independent events (like smelling the treat and solving the maze) gives the overall probability of both events happening together.

Rocco and Bliff are two koala bears participating in a series of animal behaviour tests. They each have 10 min to solve the maze. Rocco has an 85% probability of succeeding if he can smell the eucalyptus treat at the other end. He can smell the treat 60% of the time. Bliff has a 70% chance of smelling the treat, but when he does, he can solve the maze only 75% of the time. Neither bear will try to solve the maze unless he smells the eucalyptus. Determine which koala bear is more likely to enjoy a tasty treat on any given trial.

Explore how to calculate probabilities for Rocco and Bliff, two koala bears, solving a maze based on conditional probability. This problem covers key concepts like the law of total probability and provides insights into their likelihood of success in animal behaviour tests. Suitable for Grades 9-12 students.

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