Find each uniform continuous probability.

a. P(X < 10) for U (0, 50)

b. P(X > 500) for U (0, 1,000)

c. P(25< X < 45) for U(15, 65)

a. 1/5

b. 1/2
c. 2/5
(if I am interpreting the notation correctly)

a. 25

b. 500
c. .4

a. To find the probability P(X < 10) for U(0,50), we need to calculate the relative length of the interval (0, 10) within the total interval (0, 50).

Since the interval (0, 10) has a length of 10, and the total interval (0, 50) has a length of 50, the relative length is 10/50 = 1/5.

Therefore, the probability P(X < 10) is 1/5 or 0.2.

b. To find the probability P(X > 500) for U(0, 1000), we need to calculate the relative length of the interval (500, 1000) within the total interval (0, 1000).

Since the interval (500, 1000) has a length of 500, and the total interval (0, 1000) has a length of 1000, the relative length is 500/1000 = 1/2.

Therefore, the probability P(X > 500) is 1/2 or 0.5.

c. To find the probability P(25 < X < 45) for U(15, 65), we need to calculate the relative length of the interval (25, 45) within the total interval (15, 65).

Since the interval (25, 45) has a length of 20, and the total interval (15, 65) has a length of 50, the relative length is 20/50 = 2/5.

Therefore, the probability P(25 < X < 45) is 2/5 or 0.4.

To find each uniform continuous probability, we need to calculate the probability density function (PDF) of the uniform distribution and integrate it over the given interval.

The PDF of a uniform distribution is a constant function within the interval and zero outside of it.

a. P(X < 10) for U (0, 50)

To find this probability, we need to integrate the PDF over the interval (0, 10).

Step 1: Calculate the constant value of the PDF.
For a uniform distribution, the constant value is equal to 1 divided by the range of the distribution.
The range of the distribution is (50 - 0) = 50.
So, the constant value of the PDF is 1/50.

Step 2: Integrate the PDF over the interval (0, 10).
∫(0 to 10) (1/50) dx
= [x/50] (0 to 10)
= (10/50) - (0/50)
= 10/50
= 1/5
= 0.2

Therefore, P(X < 10) for U (0, 50) is 0.2.

b. P(X > 500) for U (0, 1,000)

To find this probability, we need to integrate the PDF over the interval (500, 1,000).

Step 1: Calculate the constant value of the PDF.
For a uniform distribution, the constant value is equal to 1 divided by the range of the distribution.
The range of the distribution is (1,000 - 0) = 1,000.
So, the constant value of the PDF is 1/1,000.

Step 2: Integrate the PDF over the interval (500, 1,000).
∫(500 to 1,000) (1/1,000) dx
= [x/1,000] (500 to 1,000)
= (1,000/1,000) - (500/1,000)
= 1 - 0.5
= 0.5

Therefore, P(X > 500) for U (0, 1,000) is 0.5.

c. P(25 < X < 45) for U (15, 65)

To find this probability, we need to integrate the PDF over the interval (25, 45).

Step 1: Calculate the constant value of the PDF.
For a uniform distribution, the constant value is equal to 1 divided by the range of the distribution.
The range of the distribution is (65 - 15) = 50.
So, the constant value of the PDF is 1/50.

Step 2: Integrate the PDF over the interval (25, 45).
∫(25 to 45) (1/50) dx
= [x/50] (25 to 45)
= (45/50) - (25/50)
= 0.9 - 0.5
= 0.4

Therefore, P(25 < X < 45) for U (15, 65) is 0.4.