find the probability

Question

the amount of milk produced each day by  a herd of cows is uniformly distributed between 44.6 gallons and 58.8 gallons. find the probability that the amount of milk produced is between 45.9 gallons and 50.3 gallons.

Answer 1

A uniform distribution means that the probability is constant between 44.6 and 58.8, that is  

 

P(x) = K  when 44.6 ≤ x ≤ 58.8

P(x) = 0  everywhere else  

 

If you graph P(x) you will see it forms a rectangle of height K between 44.6 and 58.8.  Since the total of all the probabilities is the area of this rectangle and must be equal to 1 you can figure out what K is by multiplying the length and width of the rectangle together and setting it equal to 1  

 

(58.8 - 44.6) * K = 1

14.2 * K = 1

K = 1/14.2  

 

The probability that the amount of milk produced is between 45.9 and 50.3 gallons is given by the area of the rectangle bounded to the left by 45.9, the right by 50.3, the top by K, and the bottom by 0.  This area is  

 

(50.3 - 45.9) * K = 4.4 / 14.2 ~ 0.31  

 

Once you get used to doing calculations like this you will realize that this is just the length of the desired interval (50.3 - 45.9) divided by the length of the entire interval (58.8 - 44.6), but I wanted to go thru all the steps so you could understand what was going on.