Question

While conducting experiments, a marine biologist selects water depths from a uniformly distributed collection that vary between 2.00 m and 7.00 m. What is the probability that a randomly selected depth is between 2.25 m and 5.00 m?

Answer 1

The probability that a randomly selected depth is between 2.25 m and 5.00 m is 0.55.

Explanation:

Let the random variable X denote the water depths.

As the variable water depths is continuous variable, the random variable X follows a continuous Uniform distribution with parameters a = 2.00 m and b = 7.00 m.

The probability density function of X is:

`f_(X)(x)=(1)/(b-a);\ a<X<b`

Compute the probability that a randomly selected depth is between 2.25 m and 5.00 m as follows:

`P(2.25<X<5.00)=\int\limits^(5.00)_(2.25) {(1)/(7.00-2.00) \, dx`

`=(1)/(5.00)\int\limits^(5.00)_(2.25) {1} \, dx\\\\=0.20* [x]^(5.00)_(2.25) \\\\=0.20* (5.00-2.25)\\\\=0.55`

Thus, the probability that a randomly selected depth is between 2.25 m and 5.00 m is 0.55.