Chapter 10.3, Problem 26E

To determine

To write:example of Bayes’ Theorem using the two way table.

Answer to Problem 26E

A survey of students’ favorite tea time snacks shows the following data.

		Favorite snacks		Total
		Nachos	Cakes
Gender	Female	$20$	$25$	$45$
	Male	$25$	$20$	$45$
	Total	$45$	$45$	$90$

Explanation of Solution

Given: Bayes’ Theorem is given by

  $P\left(A|B\right)=\frac{P\left(B|A\right)\cdot P\left(A\right)}{P\left(B\right)}$

Calculation:

A survey of students’ favorite tea time snacks shows the following data.

		Favorite snacks		Total
		Nachos	Cakes
Gender	Female	$20$	$25$	$45$
	Male	$25$	$20$	$45$
	Total	$45$	$45$	$90$

Table $10.3-26E$ survey results.

Each entry in the table is joint frequency and the sums of rows and columns are called marginal frequencies. Assume, event $A$ denote male and $B$ denote cakes.

Substitute the values in the right hand side of Bayes’ theorem formula and compute the result. The required probability is as follows:

  $\begin{array}{l}\frac{P\left(B|A\right)\cdot P\left(A\right)}{P\left(B\right)}\\ =\frac{\left(20/45\right)\left(45/90\right)}{\left(45/90\right)}\\ =\frac{20}{45}\\ =P\left(A/B\right)\end{array}$

Hence.Proved.

Conclusion:

A survey of students’ favorite tea time snacks shows the following data.

		Favorite snacks		Total
		Nachos	Cakes
Gender	Female	$20$	$25$	$45$
	Male	$25$	$20$	$45$
	Total	$45$	$45$	$90$