Question

The speed of cars passing through the intersection of Blossom Hill Road and the Almaden Expressway varies from 14 to 35 mph and is uniformly distributed. None of the cars travel over 35 mph through the intersection.Required:a. What is the probability that the speed of a car is at most 26 mph?b. What is the probability that the speed of a car is between 18 and 22 mph?c. What is the probability that the speed of a car is below 19 given that it is below 56 mph?d. What is the probability that the speed of a car is exactly 19 or 56 mph

Answer 1

a)
`P(14\leq X \leq 26) = (12)/(21)`

b)
`P(18\leq X \leq 22) = (4)/(21)`

c)
`P(X\leq 19) = P(14\leq X \leq 19) = (5)/(21)`

d)
`P(\{X=19\}\cup\{X=56\}) = 0`

Explanation:

Let X be the speed of a car. So, we know that X is distributed uniformly over the interval [14,35]. Recall that if X is distributed uniformly over the set [a,b] then the density function of X is

. Using this function we have that

`\text{P}(a \leq X\leq b) = \int_(a)^(b) f(x) dx`. In this case

`P(a\leq X \leq b = \int_(a)^(b) (1)/(21)dx = (b-a)/(21)` when a,b are in [14,35].

a).We are asked for
`P(X\leq 26) = P(14\leq X \leq 26) = (26-14)/(21)=(12)/(21)`

b)
`P(18\leq X \leq 22) = (4)/(21)`

c) Note that since 35 < 56 it is always true that the speed is under 56. So, the probability that the speed of the car is less than 19 given that the speed is below 56 is the same as having the probability that the speed is less than 19. That is

`P(X\leq 19) = P(14\leq X \leq 19) = (5)/(21)`

d) It is impossible that the speed is 19 and 56 at the same time, so the probability that the speed is 19 and 56 is 0. Since the speed is at most 35, it is impossible that the speed is 56. So the probability of having that the speed is 56 is 0. Recall from the definition that

`P(X=19) = P(19\leq X \leq 19) = (19-19)/(21) = 0`

So, using the following formula for events A,B

`P(A\cup B) = P(A)+P(B) - P(A\cap B)`

then,

`P(\{X=19\}\cup\{X=56\}) = 0`