Question

Body image A random sample of $1200$U.S. college students was asked, “What is your perception of your own body? Do you feel that you are overweight, underweight, or about right?” The two-way table below summarizes the data on perceived body image by gender.

Suppose we randomly select one of the survey respondents.

a. Given that the person perceived his or her body image as about right, what’s the

probability that the person is female?

b. If the person selected is female, what’s the probability that she did not perceive her body image as overweight?

Short answer

a. Probability for female student perceived her body image as about right is $0.6550$.

b. Probability for female student did not perceive her body image as overweight is$0.7855$.

Step-by-step solution

Given Information

It is given that:

Assuming:

A: About right

O: Overweight

U: Underweight

F: Female

Calculating probability for the female student perceived her body image as about right.

As per Conditional Probability:

$P(\mathrm{B}\mid \mathrm{A})=\frac{P(\mathrm{A}\cap \mathrm{B})}{P(\mathrm{A})}=\frac{P(\mathrm{A}\text{and}\mathrm{B})}{P(\mathrm{A})}$

Information for $1200$college students is given.

Possible outcomes are $1200$

From table, $85/1200$college students thinks that their body image is about right.

$\Rightarrow P(\mathrm{A})=\frac{\text{Number of favourable outcomes}}{\text{Number of possible outcomes}}=\frac{855}{1200}$

Also, $560/1200$female students think that their body image is about right.

$\Rightarrow P(\mathrm{A}\text{and}\mathrm{F})=\frac{\text{Number of favourableoutcomes}}{\text{Number of possible outcomes}}=\frac{560}{1200}$

Using Conditional Probability:

$P(\mathrm{A}\mid \mathrm{F})=\frac{P\left(\text{A and}\mathrm{F}\right)}{P(\mathrm{A})}=\frac{\frac{560}{1200}}{\frac{855}{1200}}=\frac{560}{855}=\frac{112}{171}\approx 0.6550=65.50\%$

It implies that $65.50\%$of students are of the view that their body image as about right are female students and the probability is$0.6550$

Calculating probability for the female student did not perceive her body image as overweight.

As per complement rule,

$P\left({\mathrm{A}}^{\mathrm{c}}\right)=P(not\mathrm{A})=1-P(\mathrm{A})$

According to conditional probability

$P(\mathrm{B}\mid \mathrm{A})=\frac{P(\mathrm{A}\cap \mathrm{B})}{P(\mathrm{A})}=\frac{P(\mathrm{A}\text{and}\mathrm{B})}{P(\mathrm{A})}$

In table, $760/1200$college students are females.

$\Rightarrow P(\mathrm{F})=\frac{\text{Number of favourableoutcomes}}{\text{Number of possibleoutcomes}}=\frac{760}{1200}$

Also, $163/1200$female students think that their body weight is about overweight.

$\Rightarrow P\left(\mathrm{O}\text{and}\mathrm{F}\right)=\frac{\text{Number of favourableoutcomes}}{\text{Number of possible outcomes}}=\frac{163}{1200}$

Using Conditional Probability

$P(\mathrm{O}\mid \mathrm{F})=\frac{P\left(\mathrm{O}\text{and}\mathrm{F}\right)}{P(\mathrm{F})}=\frac{\frac{163}{1200}}{\frac{770}{1200}}=\frac{163}{760}$

Using Complement Rule $P\left((\mathrm{O}\mid \mathrm{F}{)}^{\mathrm{c}}\right)=1-P(\mathrm{O}\mid \mathrm{F})$

$=1-\frac{163}{760}$

$=\frac{597}{760}\approx 0.7855=78.55\%$

Hence, $78.55\%$do not perceive their body image as overweight are female students and the probability is$0.7855$