Probability and Statistics for Engineering and Science, 9th Edition
Authors: Jay L. Devore
ISBN-13: 978-1305251809
See our solution for Question 59E from Chapter 2 from Devore's Probability and Statistics for Engineering and Science.
Problem 59E
Chapter:
Problem:
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 30% fill their tanks (event B). Of those customers using plus, 60% fill their tanks, whereas of those using premium, 50% fill their tanks. a. What is the probability that the next customer will request plus gas and fill the tank (A2∩B)? 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 regular gas is requested? Plus? Premium?
Step-by-Step Solution
Step 1
Probabilities of customers using regular gas:
$P\left( {{A_1}} \right) = 40\% \, = 0.4$
Probabilities of customers using plus gas
$P\left( {{A_2}} \right) = 35\% \, = 0.35$
Probabilities of customers using premium gas
$P\left( {{A_3}} \right) = 25\% \, = 0.25$
We are also given with conditional probabilities of full gas tank: \[\begin{array}{l}P\left( {B|{A_1}} \right) = 30\% = 0.3\\P\left( {B|{A_2}} \right) = 60\% = 0.6\\P\left( {B|{A_3}} \right) = 50\% = 0.5\end{array}\]
Step 2: (a)
The probability that next customer will requires plus gas and fill the tank is:\[\begin{array}{l}P\left( {{A_2} \cap B} \right) = P\left( {{A_2}} \right) \times P\left( {B|{A_2}} \right)\\ = 0.35 \times 0.60\\ = 0.21\end{array}\]Therefore, \[P\left( {{A_2} \cap B} \right) = 0.21\]
Step 3: (b)
The probability of next customer filling the tank is: \[\begin{array}{l}P\left( B \right) = P\left( {{A_1}} \right)P\left( {B|{A_1}} \right) + P\left( {{A_2}} \right)P\left( {B|{A_2}} \right) + P\left( {{A_3}} \right)P\left( {B|{A_3}} \right)\\ = \left( {0.4 \times 0.3} \right) + \left( {0.35 \times 0.6} \right) + \left( {0.25 \times 0.25} \right)\\ = 0.455\end{array}\]Therefore, \[P\left( B \right) = 0.455\]
Step 4: (c)
If the next customer fills the tank, probability of requesting regular gas is\[\begin{array}{l}P\left( {{A_1}|B} \right) = \dfrac{{P\left( {{A_1}} \right)P\left( {B|{A_1}} \right)}}{{P\left( B \right)}}\\ = \dfrac{{0.4 \times 0.3}}{{0.455}}\\ = 0.264\end{array}\]If the next customer fills the tank, probability of requesting plus gas is\[\begin{array}{l}P\left( {{A_2}|B} \right) = \dfrac{{P\left( {{A_2}} \right)P\left( {B|{A_2}} \right)}}{{P\left( B \right)}}\\ = \dfrac{{0.35 \times 0.6}}{{0.455}}\\ = 0.462\end{array}\]If the next customer fills the tank, probability of requesting premium gas is\[\begin{array}{l}P\left( {{A_3}|B} \right) = \dfrac{{P\left( {{A_3}} \right)P\left( {B|{A_3}} \right)}}{{P\left( B \right)}}\\ = \dfrac{{0.35 \times 0.5}}{{0.455}}\\ = 0.274\end{array}\]
Probabilities of customers using regular gas:
$P\left( {{A_1}} \right) = 40\% \, = 0.4$
Probabilities of customers using plus gas
$P\left( {{A_2}} \right) = 35\% \, = 0.35$
Probabilities of customers using premium gas
$P\left( {{A_3}} \right) = 25\% \, = 0.25$
We are also given with conditional probabilities of full gas tank: \[\begin{array}{l}P\left( {B|{A_1}} \right) = 30\% = 0.3\\P\left( {B|{A_2}} \right) = 60\% = 0.6\\P\left( {B|{A_3}} \right) = 50\% = 0.5\end{array}\]
Step 2: (a)
The probability that next customer will requires plus gas and fill the tank is:\[\begin{array}{l}P\left( {{A_2} \cap B} \right) = P\left( {{A_2}} \right) \times P\left( {B|{A_2}} \right)\\ = 0.35 \times 0.60\\ = 0.21\end{array}\]Therefore, \[P\left( {{A_2} \cap B} \right) = 0.21\]
Step 3: (b)
The probability of next customer filling the tank is: \[\begin{array}{l}P\left( B \right) = P\left( {{A_1}} \right)P\left( {B|{A_1}} \right) + P\left( {{A_2}} \right)P\left( {B|{A_2}} \right) + P\left( {{A_3}} \right)P\left( {B|{A_3}} \right)\\ = \left( {0.4 \times 0.3} \right) + \left( {0.35 \times 0.6} \right) + \left( {0.25 \times 0.25} \right)\\ = 0.455\end{array}\]Therefore, \[P\left( B \right) = 0.455\]
Step 4: (c)
If the next customer fills the tank, probability of requesting regular gas is\[\begin{array}{l}P\left( {{A_1}|B} \right) = \dfrac{{P\left( {{A_1}} \right)P\left( {B|{A_1}} \right)}}{{P\left( B \right)}}\\ = \dfrac{{0.4 \times 0.3}}{{0.455}}\\ = 0.264\end{array}\]If the next customer fills the tank, probability of requesting plus gas is\[\begin{array}{l}P\left( {{A_2}|B} \right) = \dfrac{{P\left( {{A_2}} \right)P\left( {B|{A_2}} \right)}}{{P\left( B \right)}}\\ = \dfrac{{0.35 \times 0.6}}{{0.455}}\\ = 0.462\end{array}\]If the next customer fills the tank, probability of requesting premium gas is\[\begin{array}{l}P\left( {{A_3}|B} \right) = \dfrac{{P\left( {{A_3}} \right)P\left( {B|{A_3}} \right)}}{{P\left( B \right)}}\\ = \dfrac{{0.35 \times 0.5}}{{0.455}}\\ = 0.274\end{array}\]