Question

At a gas station, 40% of the customers use regular gas (A1), 35% use mid-grade gas (A2), and 25% use premium gas (A3). Of those customers using regular gas, only 30% fill their tanks (event B). Of those customers using mid-grade gas, 60% fill their tanks, whereas of those using premium, 50% fill their tanks.

a. What is the probability that the next customer will requestplus gas and fill their tank ?
b. What is the probability that the next customer fills the tank ?
c. If the next customer fills the tank, what is the probability that the regular gas is requested?

Answer 1

Answer:

(a) 0.12

(b) 0.455

(c) 0.264

Step-by-step explanation:

We are given that Probability of customers using regular gas,P(A1) = 0.4

Probability of customers using mid-grade gas,P(A2) = 0.35

Probability of customers using premium gas,P(A3) = 0.25

Let Event B = Customers fill their tank

So, P(B/A1) = 0.3 {This means that Probability that customers fill their tank   given they are using regular gas is 30%}

Similarly P(B/A2) = 0.6     and     P(B/A3) = 0.5

Now, In general P(A/B) = $\frac{P(A\bigcap B)}{P(B)}$ or $P(A\bigcap B)$ = P(A/B) * P(B) .

(a) Probability that the next customer will request regular gas and fill their tank = $P(A1\bigcap B)$ {For happening of both events we use intersection sign}

       = P(B/A1) * P(A1) [Note: $P(A1\bigcap B)$ is same as $P(B\bigcap A1)$  )

       = 0.3 * 0.4 = 0.12

(b)  Probability that the next customer fills the tank is given by the cases:

• Customer uses regular gas and fills the tank  - $P(A3\bigcap B)$

• Customer uses mid-grade gas and fills the tank - $P(A2\bigcap B)$

• Customer uses regular gas and fills the tank -  $P(A1\bigcap B)$

So, P(B) =  $P(A1\bigcap B)$ + $P(A2\bigcap B)$ + $P(A3\bigcap B)$

             = P(B/A1) * P(A1) + P(B/A2) * P(A2) + P(B/A3) * P(A3)

             = 0.3 * 0.4 + 0.6 * 0.35 + 0.5 * 0.25 = 0.455

(c) If the next customer fills the tank, probability that the regular gas is requested is given by the expression P(A1/B) because this states the Probability of requesting regular gas given customer has filled the tank.

So, P(A1/B) = $\frac{P(A1\bigcap B)}{P(B)}$ = $\frac{0.12}{0.455}$ = 0.264