Chapter 5.3, Problem 83E

To determine

To find: the probability that the customer paid with a credit card. Also, to draw the tree diagram to represent the situation.

Answer to Problem 83E

Theprobability that the customer paid with a credit card is $0.2952\%$ .

Explanation of Solution

Given:

88% of the automobiles filled their vehicles with regular gasoline, 2% purchase midgrade gas and 10% bought premium gas.

28% is paid with credit card on regular gas, 34% is paid on midgrade gas, and 42% on premium gas.

Concept used:

Definition of conditional probability:

  $P\left(A\cap B\right)=P\left(A/B\right)\times P\left(B\right)$

Calculation:

According to the question, the tree diagram is:

  

Now, to findthe probability of paying with credit card for each type of gasoline.

  $\begin{array}{l}P\left(\text{Regular}\cap \text{Credit card}\right)=P\left(\text{Credit card/Regular}\right)\times P\left(\text{Regular}\right)=88\%\times 28\%=0.88\times 0.28=0.2464\\ P\left(\text{Midgrade}\cap \text{Credit card}\right)=P\left(\text{Credit card/Midgrade}\right)\times P\left(\text{Midgrade}\right)=2\%\times 34\%=0.02\times 0.34=0.0068\\ P\left(\text{Premium}\cap \text{Credit card}\right)=P\left(\text{Credit card/Premium}\right)\times P\left(\text{Premium}\right)=10\%\times 42\%=0.10\times 0.42=0.042\end{array}$

Add the corresponding probabilities:

  $\begin{array}{l}P\left(\text{credit card}\right)\\ =0.2464+0.0068+0.042\\ =0.2952\%\end{array}$

Hence,the probability that the customer paid with a credit card is $0.2952\%$ .