Question

the amount of gas in sarahs car is uniformly distributed between 1 and 16 gallons. Calculate the probability that the amount of gas is exactly 7 gallons

Answer 1

Answer:

The probability that the amount of gas in Sarah's car is exactly 7 gallons is 0.067.

Step-by-step explanation:

Let the random variable X represent the amount of gas in Sarah's car.

It is provided that $X\sim Unif(1, 16)$.

The amount of gas in a car is a continuous variable.

So, the random variable X follows a continuous uniform distribution.

Then the probability density function of X is:

$f_{X}(x)=\frac{1}{b-a};\ a$

For a continuous probability distribution the probability at an exact point is 0.

So, to compute the probability that the amount of gas in Sarah's car is exactly 7 gallons use continuity correction on both sides:

P (X = 7) = P (7 - 0.5 < X < 7 + 0.5)

              = P (6.5 < X < 7.5)

              $=\int\limits^{7.5}_{6.5} {\frac{1}{16-1}} \, dx \\\\=\frac{1}{15}\times |x|^{7.5}_{6.5}\\\\=\frac{1}{15}\times (7.5-6.5)\\\\=\frac{1}{15}\\\\=0.0666667\\\\\approx 0.067$

Thus, the probability that the amount of gas in Sarah's car is exactly 7 gallons is 0.067.