How do you calculate the probability of an event occurring in a subinterval for a continuous uniform distribution?

Conditions

• The distribution is continuous and uniform over the main interval.
• The target subinterval lies completely inside the main interval.
• Probability is represented by the area under the density graph.

Reasoning, step by step

1. Identify the lower and upper endpoints of the target subinterval.
2. Calculate the base of the rectangle by subtracting the lower endpoint from the upper endpoint.
3. Identify the constant height of the probability density function.
4. Multiply the base by the height to calculate the area, which equals the probability.

Example

For the bus waiting time uniformly distributed on [0, 60], the probability of waiting between 5 and 20 minutes is found by calculating the area of the sub-rectangle. The base is $20 - 5 = 15$, and the height is $\frac{1}{60}$. The probability is $\frac{15}{60} = 0.25$, or 25%.

Common misconceptions

• Assuming the probability is simply the difference between the endpoints without multiplying by the density height.
• Confusing the probability of an exact single point (which is zero in a continuous distribution) with the probability of an interval.

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