The weekly amount of gas sold at a self serve gas station has a uniform distribution on the interval 5000 to 15,000 gallons. If the dealer begins the week with 12,625 gallons in his supply tank what is the probability that he will run out of gas before the end of the week

To find the probability that the dealer will run out of gas before the end of the week, we need to determine the proportion of the uniform distribution that is less than the amount of gas in the supply tank.

A uniform distribution is characterized by a constant probability density function (PDF) over a specified interval. In this case, the interval is from 5000 to 15,000 gallons. The PDF is given by:

f(x) = 1 / (b - a)

where a is the lower bound of the interval (5000 gallons) and b is the upper bound of the interval (15000 gallons).

To find the probability that the dealer will run out of gas before the end of the week, we need to calculate the cumulative distribution function (CDF) at the dealer's starting supply tank level.

The CDF is defined as the integral of the PDF from the lower bound to some value x:

F(x) = ∫[a, x] f(t) dt

In this case, we want to calculate F(12625).

Using the formula for the PDF and integrating, we get:

F(x) = (x - a) / (b - a)

Replacing x with 12625, a with 5000, and b with 15000, we have:

F(12625) = (12625 - 5000) / (15000 - 5000) = 7625 / 10000 = 0.7625

Thus, the probability that the dealer will run out of gas before the end of the week is 0.7625, or 76.25%.

Remember that this probability assumes a continuous distribution and does not take into account any other factors such as daily sales patterns or variations in demand.