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18 May, 01:34

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 20% fill their tanks (event B). Of those customers using plus, 40% fill their tanks, whereas of those using premium, 50% fill their tanks.

Consider the following information on credit card usage.

1. 70% of all regular fill-up customers use a credit card.

2. 50% of all regular non-fill-up customers use a credit card.

3. 60% of all plus fill-up customers use a credit card.

4. 50% of all plus non-fill-up customers use a credit card.

5. 50% of all premium fill-up customers use a credit card.

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 thetank?

c) If the next customer fills the tank, what is the probabilitythat the regular gas is requested? Plus? Premium

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Answers (1)
  1. 18 May, 01:57
    0
    a.)

    P (A₂ ∩ B) = P (B | A₂) * P (A₂)

    P (A₂ ∩ B) = 0.40 * 0.35

    P (A₂ ∩ B) = 0.14

    b.)

    P (B) = P (A₁ ∩ B) + P (A₂ ∩ B) + P (A₃ ∩ B)

    P (B) = 0.08 + 0.14 + 0.125

    P (B) = 0.345

    c.)

    For regular gas:

    P (A₁ | B) = P (A₁ ∩ B) / P (B)

    P (A₁ | B) = 0.08 / 0.345

    P (A₁ | B) = 0.232

    For plus gas:

    P (A₂ | B) = P (A₂ ∩ B) / P (B)

    P (A₂ | B) = 0.14 / 0.345

    P (A₂ | B) = 0.406

    For premium gas:

    P (A₃ | B) = P (A₃ ∩ B) / P (B)

    P (A₃ | B) = 0.125 / 0.345

    P (A₃ | B) = 0.362

    Step-by-step explanation:

    We are given the following information

    40% of the customers use regular gas (A2)

    P (A₁) = 0.40

    35% use plus gas (A2)

    P (A₂) = 0.35

    25% use premium (A3)

    P (A₃) = 0.25

    Of those customers using regular gas, only 20% fill their tanks (event B).

    P (B | A₁) = 0.20

    Of those customers using plus, 40% fill their tanks

    P (B | A₂) = 0.40

    Whereas of those using premium, 50% fill their tanks.

    P (B | A₃) = 0.5

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

    We are asked to find P (A₂ ∩ B) = ?

    Recall that Multiplicative law of probability is given by

    P (A₂ ∩ B) = P (B | A₂) * P (A₂)

    P (A₂ ∩ B) = 0.40 * 0.35

    P (A₂ ∩ B) = 0.14

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

    We are asked to find P (B) = ?

    P (B) = P (A₁ ∩ B) + P (A₂ ∩ B) + P (A₃ ∩ B)

    P (A₂ ∩ B) is already calculated, we need to calculate

    P (A₁ ∩ B) and P (A₃ ∩ B)

    So,

    P (A₁ ∩ B) = P (B | A₁) * P (A₁)

    P (A₁ ∩ B) = 0.20 * 0.40

    P (A₁ ∩ B) = 0.08

    P (A₃ ∩ B) = P (B | A₃) * P (A₃)

    P (A₃ ∩ B) = 0.50 * 0.25

    P (A₃ ∩ B) = 0.125

    Finally,

    P (B) = P (A₁ ∩ B) + P (A₂ ∩ B) + P (A₃ ∩ B)

    P (B) = 0.08 + 0.14 + 0.125

    P (B) = 0.345

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

    For regular gas:

    P (A₁ | B) = P (A₁ ∩ B) / P (B)

    P (A₁ | B) = 0.08 / 0.345

    P (A₁ | B) = 0.232

    For plus gas:

    P (A₂ | B) = P (A₂ ∩ B) / P (B)

    P (A₂ | B) = 0.14 / 0.345

    P (A₂ | B) = 0.406

    For premium gas:

    P (A₃ | B) = P (A₃ ∩ B) / P (B)

    P (A₃ | B) = 0.125 / 0.345

    P (A₃ | B) = 0.362
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