Table relates the weights and heights of a group of individuals participating in an observational study.

a. Find the total for each row and column

b. Find the probability that a randomly chosen individual from this group is Tall.

c. Find the probability that a randomly chosen individual from this group is Obese and Tall.

d. Find the probability that a randomly chosen individual from this group is Tall given that the individual is Obese.

e. Find the probability that a randomly chosen individual from this group is Obese given that the individual is Tall.

f. Find the probability a randomly chosen individual from this group is Tall and Underweight.

g. Are the events obese and Tall independent?

The table relates the weights and heights of a group of individuals participating in an observational study.

The total for each row and column are :

The total for each row and column are :

The probability that a randomly chosen individual from this group is Tall is calculated as:

$$\begin{gathered} P\left(\text{Tall}\right)=\frac{\text{Individual in Tall group}}{\text{Total Individual}} \\ =\frac{50}{205} \\ =0.244 \end{gathered}$$

The probability that a randomly chosen individual from this group is Obese and Tall is calculated as:

$$\begin{gathered} P\left(\text{Obese and Tall}\right)=\frac{\text{Individual in Obese and Tall group}}{\text{Total Individual}} \\ =\frac{18}{205} \\ =0.088 \end{gathered}$$

The probability that a randomly chosen individual from this group is Tall given that the individual is Obese is calculated as:

$$\begin{gathered} P(\text{Tall}\mid \text{Obese})=\frac{\text{Individual in Tall group given Obese}}{\text{Total Obese}} \\ =\frac{18}{60} \\ =0.3 \end{gathered}$$

The probability that a randomly chosen individual from this group is Obese given that the individual is Tall is calculated as :

$$\begin{gathered} P(\text{Obese}\mid \text{Tall})=\frac{\text{Individual in Obese group given Tall}}{\text{Total Tall}} \\ =\frac{18}{50} \\ =0.36 \end{gathered}$$

The probability a randomly chosen individual from this group is Tall and Underweight is calculated as:

$$\begin{gathered} P\left(\text{Tall and Underweight}\right)=\frac{\text{Individual in Tall and Underweight group}}{\text{Total Individual}} \\ =\frac{12}{205} \\ =0.059 \end{gathered}$$

To prove the independency of events M and H, we must prove the following three conditions:

$$\begin{gathered} a.P(Tall\backslash Obese)=P(Obese) \\ b.P(Obese\backslash Tall)=P(Tall) \\ c.P(ObeseANDTall)=P(Obese)P(Tall) \end{gathered}$$

We have

$$\begin{gathered} P(\text{Tall}\mid \text{Obese})=0.3 \\ \text{And}P\left(\text{Obese}\right)=0.088 \\ \text{Since}P(\text{Tall}\mid \text{Obese})\ne P\left(\text{Obese}\right) \end{gathered}$$

As a result, the first requirement of independence is not met.

As a result, the occurrences are not independent.