Question

In a large city, one in ten fire hydrants are in need of repair. If a crew examines 100 fire hydrants in a week, what is the probability they will find nine or fewer fire hydrants that need repair? Use the normal distribution to approximate the binomial distribution.

What is the probability of finding nine or fewer fire hydrants that need repair?
a) 0.25
b) 0.50
c) 0.75
d) 0.90

Answer 1

To find the probability of finding nine or fewer fire hydrants in need of repair out of 100 examined, we can use the binomial distribution and approximate it with the normal distribution. Using the appropriate formula and calculations, the probability is approximately 0.3693.

Step-by-step explanation:

To find the probability of finding nine or fewer fire hydrants that need repair, we can use the binomial distribution and approximate it with the normal distribution. The binomial distribution can be approximated by the normal distribution when np and n(1-p) are both greater than or equal to 5, where n is the number of trials and p is the probability of success in each trial.

In this case, n = 100 (number of fire hydrants examined) and p = 0.1 (probability that a fire hydrant needs repair). Therefore, np = 100 * 0.1 = 10 and n(1-p) = 100 * (1-0.1) = 90.

Using the normal distribution, we need to find the z-score for nine hydrants (x = 9) using the formula:

z = (x - np) / sqrt(np(1-p))

Substituting the values, we get:

z = (9 - 10) / sqrt(10 * (1-0.1))

z = -1 / sqrt(9)

z ≈ -1 / 3

Looking up the z-score in the standard normal distribution table, we find that the area to the left of -1/3 (or the probability of finding nine or fewer hydrants in need of repair) is approximately 0.3693.

Therefore, the probability of finding nine or fewer fire hydrants that need repair is approximately 0.3693.