Respuesta :
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) = [tex]\frac{P(A\bigcap B)}{P(B)}[/tex] or [tex]P(A\bigcap B)[/tex] = P(A/B) * P(B) .
(a) Probability that the next customer will request regular gas and fill their tank = [tex]P(A1\bigcap B)[/tex] {For happening of both events we use intersection sign}
= P(B/A1) * P(A1) [Note: [tex]P(A1\bigcap B)[/tex] is same as [tex]P(B\bigcap A1)[/tex] )
= 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 - [tex]P(A3\bigcap B)[/tex]
- Customer uses mid-grade gas and fills the tank - [tex]P(A2\bigcap B)[/tex]
- Customer uses regular gas and fills the tank - [tex]P(A1\bigcap B)[/tex]
So, P(B) = [tex]P(A1\bigcap B)[/tex] + [tex]P(A2\bigcap B)[/tex] + [tex]P(A3\bigcap B)[/tex]
= 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) = [tex]\frac{P(A1\bigcap B)}{P(B)}[/tex] = [tex]\frac{0.12}{0.455}[/tex] = 0.264
