Chapter 10.1, Problem 57E

Solution

(a).

To make: A table showing the approximate probability distribution.

The complete table will be:

  $\begin{array}{cc}\text{Type of Bagel}& \text{Number Sold}\\ \text{Plain}& 0.37\\ \text{Onion}& 0.12\\ \text{Rye}& 0.11\\ \text{Cinnamon Raisin}& 0.25\\ \text{Sourdough}& 0.15\end{array}$

Given information:

In real applications, it is often necessary to approximate the probabilities of the various outcomes of an experiment by performing the experiment a large number of times and recording the results. Barney’s Bread Basket offers five different kinds of bagels. Barney records the sales of the first $500$ bagels in a given week in the table shown below:

  $\begin{array}{cc}\text{Type of Bagel}& \text{Number Sold}\\ \text{Plain}& 185\\ \text{Onion}& 60\\ \text{Rye}& 55\\ \text{Cinnamon Raisin}& 125\\ \text{Sourdough}& 75\end{array}$

Calculation:

Divide each type of bagel by the total number sold to find their corresponding probabilities. In this case, the total number sold is:

  $185+60+55+125+75=500$

So, the completed table will be:

  $\begin{array}{cc}\text{Type of Bagel}& \text{Number Sold}\\ \text{Plain}& \frac{185}{500}=0.37\\ \text{Onion}& \frac{60}{500}=0.12\\ \text{Rye}& \frac{55}{500}=0.11\\ \text{Cinnamon Raisin}& \frac{125}{500}=0.25\\ \text{Sourdough}& \frac{75}{500}=0.15\end{array}$

(b).

To find: The probability that three customers in a row all order plain bagels.

The probability that three customers in a row all order plain bagels is $0.05063$ .

Given information:

In real applications, it is often necessary to approximate the probabilities of the various outcomes of an experiment by performing the experiment a large number of times and recording the results. Barney’s Bread Basket offers five different kinds of bagels. Barney records the sales of the first $500$ bagels in a given week in the table shown below:

  $\begin{array}{cc}\text{Type of Bagel}& \text{Number Sold}\\ \text{Plain}& 185\\ \text{Onion}& 60\\ \text{Rye}& 55\\ \text{Cinnamon Raisin}& 125\\ \text{Sourdough}& 75\end{array}$

Calculation:

Divide each type of bagel by the total number sold to find their corresponding probabilities. In this case, the total number sold is:

  $185+60+55+125+75=500$

So, the completed table will be:

  $\begin{array}{cc}\text{Type of Bagel}& \text{Number Sold}\\ \text{Plain}& \frac{185}{500}=0.37\\ \text{Onion}& \frac{60}{500}=0.12\\ \text{Rye}& \frac{55}{500}=0.11\\ \text{Cinnamon Raisin}& \frac{125}{500}=0.25\\ \text{Sourdough}& \frac{75}{500}=0.15\end{array}$

Since the events are independent, you can multiply them and find the probability.

  $0.37\cdot 0.37\cdot 0.37=0.050653$

Hence, probability that three customers in a row all order plain bagels is $0.05063$.

(c).

To explain: It is reasonable to assume that the orders of three consecutive customers actually are independent.

The probability of the order of the second customer is affected by the order of the first customer and thus the events are not independent.

Given information:

In real applications, it is often necessary to approximate the probabilities of the various outcomes of an experiment by performing the experiment a large number of times and recording the results. Barney’s Bread Basket offers five different kinds of bagels. Barney records the sales of the first $500$ bagels in a given week in the table shown below:

  $\begin{array}{cc}\text{Type of Bagel}& \text{Number Sold}\\ \text{Plain}& 185\\ \text{Onion}& 60\\ \text{Rye}& 55\\ \text{Cinnamon Raisin}& 125\\ \text{Sourdough}& 75\end{array}$

Calculation:

Divide each type of bagel by the total number sold to find their corresponding probabilities. In this case, the total number sold is:

  $185+60+55+125+75=500$

So, the completed table will be:

  $\begin{array}{cc}\text{Type of Bagel}& \text{Number Sold}\\ \text{Plain}& \frac{185}{500}=0.37\\ \text{Onion}& \frac{60}{500}=0.12\\ \text{Rye}& \frac{55}{500}=0.11\\ \text{Cinnamon Raisin}& \frac{125}{500}=0.25\\ \text{Sourdough}& \frac{75}{500}=0.15\end{array}$

Two events are independent, if the probability that one event occurs in no way affects the probability of the other event occurring.

When taking the orders of three consecutive customers, then it is likely that these customers are family or friends.

Since friends and/or family tend to have the same preferences, we are then more likely to obtain the same orders among three consecutive people than among randomly selected people.

However, this then implies that the probability of the order of the second customer is affected by the order of the first customer and thus the events are not independent.