Question

A recent survey in Spain revealed that 60% of the vehicles traveling on highways, where speed limits are posted at 120 km per hour, were exceeding the limit. Suppose you randomly record the speeds of ten vehicles traveling on 311 where the speed limit is 120 km per hour. Let X denote the number of vehicles that were exceeding the limit. a. What is the distribution of X? b. Find P(X = 10). c. Find the probability that the number of vehicles exceeding the speed limit is between 4 and 9(exclusive) d. Find the probability that two of the recorded vehicles were exceeding the speed limit

Answer 1

a)

This is a binomial distribution

n = 10 , p = 0.60

b)

Here, n = 10, p = 0.6, (1 - p) = 0.4 and x = 10
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X = 10)
P(X = 10) = 10C10 * 0.6^10 * 0.4^0
P(X = 10) = 0.006
0

c)

Here, n = 10, p = 0.6, (1 - p) = 0.4, x1 = 4 and x2 = 9.
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(4 <= X <= 9)
P(4 <= X <= 9) = (10C5 * 0.6^5 * 0.4^5) + (10C6 * 0.6^6 * 0.4^4) + (10C7 * 0.6^7 * 0.4^3) + (10C8 * 0.6^8 * 0.4^2)
P(4 <= X <= 9) = 0.2007 + 0.2508 + 0.215 + 0.1209
P(4 <= X <= 9) = 0.7874

d)
Here, n = 10, p = 0.6, (1 - p) = 0.4 and x = 2
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X = 2)
P(X = 2) = 10C2 * 0.6^2 * 0.4^8
P(X = 2) = 0.0106