Let's break down the problem step by step.

Step 1: Understanding the Distribution

The time taken by teenagers to finish a box of donuts is uniformly distributed between 1 and 6 minutes. This means that any time within this range is equally likely. The probability density function (PDF) for a uniform distribution $U(a, b)$ is given by:

$f(x) = \frac{1}{b-a}, \quad \text{for } a \leq x \leq b$

Here, $a = 1$ minute and $b = 6$ minutes. So:

$f(x) = \frac{1}{6-1} = \frac{1}{5}, \quad \text{for } 1 \leq x \leq 6$

Step 2: Event Definitions

• Let $A$ be the event that Ajay finishes the box of donuts within 4 minutes.
• Let $B$ be the event that Ajay loses the challenge, i.e., takes more than 100 seconds (which is 1.67 minutes).

We need to find the conditional probability $P(A \mid B)$, which is the probability that Ajay finishes the box within 4 minutes given that he took more than 1.67 minutes.

Step 3: Calculate Individual Probabilities

1. Probability of Event $A$: $A = \{1 \leq x \leq 4\}$.

   The probability that Ajay finishes the box within 4 minutes is: $P(A) = \int_1^4 f(x) \, dx = \int_1^4 \frac{1}{5} \, dx = \frac{4-1}{5} = \frac{3}{5}$

2. Probability of Event $B$: $B = \{x > 1.67\}$.

   The probability that Ajay took more than 1.67 minutes is: $P(B) = \int_{1.67}^6 f(x) \, dx = \int_{1.67}^6 \frac{1}{5} \, dx = \frac{6-1.67}{5} = \frac{4.33}{5}$

3. Intersection of Events $A$ and $B$: $A \cap B = \{1.67 \leq x \leq 4\}$.

   The probability that Ajay took between 1.67 and 4 minutes is: $P(A \cap B) = \int_{1.67}^4 f(x) \, dx = \int_{1.67}^4 \frac{1}{5} \, dx = \frac{4-1.67}{5} = \frac{2.33}{5}$

Step 4: Calculate the Conditional Probability

The conditional probability $P(A \mid B)$ is given by:

$P(A \mid B) = \frac{P(A \cap B)}{P(B)} = \frac{\frac{2.33}{5}}{\frac{4.33}{5}} = \frac{2.33}{4.33} \approx 0.538$

Conclusion

The probability that Ajay will finish the box of donuts within 4 minutes, given that he took more than 1.67 minutes, is approximately 0.538 or 53.8%.

Would you like to discuss this further or have any specific questions?

Follow-up Questions:

1. How would the probability change if the time range were different?
2. What if the challenge time was set to 3 minutes instead of 100 seconds?
3. Can we calculate the expected time for Ajay to finish the donuts?
4. What other real-life situations can be modeled using a uniform distribution?
5. How does the uniform distribution differ from other continuous distributions?

Tip:

Conditional probability can often provide surprising insights, especially when combined with uniform distributions, where each outcome is equally likely within a given range.