Question

At a certain gas station, 40% of the customers use regular gas (A1), 35% use plus gas (A2), and 25% use premium (A3). Of those customers using regular gas, only 10% fill their tanks (event B). Of those customers using plus, 20% fill their tanks, whereas of those using premium, 30% fill their tanks.

Required:
a. What is the probability that the next customer will request plus 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

a. The probability of the next customer requesting plus gas and filling their tank is 7%. b. The probability of the next customer filling their tank is 15%. c. If the next customer fills their tank, the probability of them requesting regular gas is 26.67%.

Step-by-step explanation:

To answer the given questions, we need to use conditional probability. Let's denote the events as follows:

A1: Customer uses regular gas

A2: Customer uses plus gas

A3: Customer uses premium gas

B: Customer fills their tank

a. We want to find the probability that the next customer will request plus gas (A2) and fill their tank (B). This can be written as P(A2 and B). We know that 35% of customers use plus gas, and out of those, 20% fill their tanks. Therefore, P(A2 and B) = P(A2) × P(B|A2) = 0.35 * 0.20 = 0.07 or 7%.

b. We want to find the probability that the next customer fills their tank (B). This can be written as P(B). To calculate this, we need to consider customers who use each type of gas and the corresponding probabilities of filling their tanks. The probability can be calculated as follows: P(B) = P(A1) × P(B|A1) + P(A2) × P(B|A2) + P(A3) × P(B|A3) = 0.4 × 0.1 + 0.35 × 0.2 + 0.25 × 0.3 = 0.15 or 15%.

c. We want to find the probability that if the next customer fills their tank (B), the regular gas is requested (A1). This can be written as P(A1|B). To calculate this, we use Bayes' theorem: P(A1|B) = P(A1) × P(B|A1) / P(B). We already have the values needed: P(A1) = 0.4, P(B|A1) = 0.1, and P(B) = 0.15. Substituting these values, we find P(A1|B) = 0.4 × 0.1 / 0.15 = 0.2667 or 26.67%.

Answer 2

Remember that for an event with a percentage probability X, the probability is given by:

P = X/100%

a) What is the probability that the next customer will request plus gas and fill their tank ?

We know that 35% of the customers use plus gas, and of these using plus, 20% fill their tank.

So the probability that the next customer uses plus gas is:

p = 35%/100% = 0.35

And the probability that the customer fills the tank (given that the customer uses plus gas) is:

q = 20%/100% = 0.2

The joint probability is just the product between the individual probabilities:

Then the probability that the next customer uses plus gas and fills their tank is:

P = 0.35*0.2 = 0.07

b) What is the probability that the next customer fills the tank?

We know that 20% of the ones that use plus gas (with a probability of 0.35) fill their tank, 10% of these that use regular gas (with a probability of 0.4) and 30% of these that use premium (with a probability of 0.25) fill their tank,

Then the probability is computed in a similar way than above, here the probability is:

P = 0.2*0.35 + 0.1*0.4 + 0.3*0.25 = 0.185

The probability that the next customer fills the tank is 0.185

c) If the next customer fills the tank, what is the probability that the regular gas is requested?

Ok, now we already know that the customer fills the tank.

The probability that a customer uses regular and fills the tank, is

p = 0.1*0.4

The probability that a customer fills the tank is computed above, this is:

P = 0.185

The probability, given that the customer fills the tank, the customer uses regular gas, is equal to the quotient between the probability that the customer fills the tank with regular and the probability that the customer fills the tank, this is:

Probability = p/P = (0.1*0.4)/(0.185) = 0.216