Answer:
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
