Question

The time it takes me to wash the dishes is uniformly distributed between 8 minutes and 17 minutes.

What is the probability that washing dishes tonight will take me between 14 and 16 minutes?

Give your answer accurate to two decimal places.

Answer 1

The time it takes to wash has the probability density function,

`P(X=x) = \begin{cases}\frac1{17-8}=\frac19&\text{for }8\le x\le 17\\0&\text{otherewise}\end{cases}`

The probability that it takes between 14 and 16 minutes to wash the dishes is given by the integral,

`\displaystyle\int_(14)^(16)P(X=x)\,\mathrm dx = \frac19\int_(14)^(16)\mathrm dx = \frac{16-14}9 = \frac29 \approx \boxed{0.22}`

If you're not familiar with calculus, the probability is equal to the area under the graph of P(X = x), which is a rectangle with height 1/9 and length 16 - 14 = 2, so the area and hence probability is 2/9 ≈ 0.22.