Question

3. Let the amount of rain on rainy days be normally distributed with a mean of 3 cm and standard deviation of 1cm.

(a) Find the probability that there is at least 2cm of rain on a randomly chosen rainy day. You can use a normal table to get the probabilities needed or use pnorm() in R. Do get a final answer that is numerical and not just a function of Φ(), the standard normal cdf.

(b) Suppose on rainy days the rate of a car accidents at a given intersection is .2 per day, and on non-rainy days, the rate of car accidents at the same intersection is .1 per day. If the probability of a rainy day tomorrow is 0.3, what is the probability that there will be no car accidents tomorrow at the intersection.

(c) Using the same information as (b), suppose there is exactly one car accident at the intersection on a particular day, but you don’t know whether it was rainy that day (maybe you are visiting Australia...). What is the probability that it was raining that day?

Answer 1

(a)

$\mu$ = 3

$\sigma$ = 1

To find P(X $\geq$ 2):

Z = (2 - 3)/1

= - 1.00

Table of Area Under Standard Normal Curve gives area = 0.3413

So,

P(X $\geq$ 2) = 0.5 + 0.3413 = 0.8413

So,

Answer is:

0.8413

(b)

Case 1:

P(Rain tomorrow AND Accident tomorrow) = P(Rain tomorrow) X P(Accident tomorrow) = 0.3 X 0.2 = 0.06

Case 2:

P(No rain tomorrow AND Accident tomorrow) = P(No rain tomorrow) X P(Accident tomorrow) = 0.7 X 0.1 = 0.07

So,

P(Accident tomorrow) = 0.06 + 0.07 = 0.13

So,

P(No accident tomorrow) - 1 - 0.13 = 0.87

So,

Answer is:

0.87

(c)

P(Raining/ Accident) = P(Raining AND Accident)/P(Accident)

                               = 0.06/0.13

                                = 0.4615

So,

Answer is:

0.4615