Question

A driver has three ways to get from one city to another. There is an 80 % chance of encountering atraffic jam given he is on route A , a 60 % chance given he is on route B , and a 30 % chance given he is onroute C. Because of other factors, such as distance and speed limits , the driver uses route A 50 % of thetime and routes B and C each 25 % of the time. If the driver calls the dispatcher to inform him that he isin a traffic jam, find the probability that he selected route A

Answer 1

To find the probability that the driver selected route A given that he called the dispatcher to inform him about a traffic jam, we use conditional probability and Bayes' theorem. The probability is found to be 8/13.

Step-by-step explanation:

To find the probability that the driver selected route A given that he called the dispatcher to inform him about a traffic jam, we need to use conditional probability. Let's use the notation P(A|D) to represent the probability of selecting route A given that there is a traffic jam. We can use Bayes' theorem to calculate this probability:

P(A|D) = P(D|A) * P(A) / P(D)

Where:

• P(D|A) is the probability of encountering a traffic jam given that the driver is on route A (0.8)
• P(A) is the probability of selecting route A (0.5)
• P(D) is the probability of encountering a traffic jam (0.5*0.8 + 0.25*0.6 + 0.25*0.3)

Now we can substitute these values into the formula and calculate the probability:

P(A|D) = (0.8 * 0.5) / (0.5*0.8 + 0.25*0.6 + 0.25*0.3) = 0.32 / 0.52 = 16/26 = 8/13

Therefore, the probability that the driver selected route A given that he called the dispatcher to inform about a traffic jam is 8/13.

Answer 2

The probability is 0.64

Step-by-step explanation:

What we want to calculate here is conditional probability.

Let P(A )= Probability of using route A = 50% = 0.5

Let P( B )= probability of using route B = 25% = 0.25

Let P (C )= probability of using route C = 25% = 0.25

Let T be the probability that he will be in a traffic Jam

The probability that he will be in a traffic Jam if he uses route A = 80%

Mathematically this is written as P( T | A) which is read as probability of T given A

so P( T | A) = 0.8

Same way for B and C which can be written as follows;

P( T | B) = 60% = 0.6

P( T | C) = 30% = 0.3

Now, what do we want to calculate?

He is in a traffic Jam, and we want to find the probability that he used route A. This means we want to find P(A) given T which can be written mathematically as P ( A | T)

We can find this using the other parameters and especially the equation below;

P ( A | T) = P(A) • P( T| A) / {P(A) • P ( T | A) + P(B)• P(T| B) + P(C) • P(T|C)

P( A | T) = (0.5 * 0.8)/ ( 0.5)(0.8) + (0.25)(0.6) + (0.25)(3) = 0.4/(0.4 + 0.75 + 0.075) = 0.4/0.625 = 0.64