Chapter 10, Problem AP3.2CPT

To determine

To find out from the options that which of the following probabilities is the largest.

Answer to Problem AP3.2CPT

Option (b) is the correct option.

Explanation of Solution

The table is given in the question. The table contains $998$ U.S. adults in total. From those U.S. adults $360$ are Democrat, $404$ are Independent and $234$ are Republican. And these are divide into categories as well which are: those which are expected to live longer, those who are expected to live about as long, those who are expected to live not so long and those who are expected to be not sure about.

Now, we have to calculate the largest probability from the options given in the question. Thus,

Probability is the number of favorable outcomes divided by the number of possible outcomes.

From the help of the table given in the question we will find the probabilities of the following:

P(Independent and Longer) $=\frac{134}{998}=0.1343$

Where we know that $134$ are the U.S. adults that are independent and are expected to live longer from the total of $998$ U.S. adults.

Now, for the other probability, we have,

P (Independent or not so long)

  $\begin{array}{l}=\frac{470}{998}\\ =0.4709\end{array}$

Where we know that $470$ are the U.S. adults that are independent and are expected to live not so long from the total of $998$ U.S. adults.

Now, for the other probability, we have,

P (Democrat| Not as long)

  $\begin{array}{l}=\frac{43}{115}\\ =0.3739\end{array}$

It is a conditional probability, where, the table contains $115$ individuals who are expected to not live as long while $43$ of the $115$ individuals who are expected to not live as long are democrat.

Now, for the other probability, we have,

P (About as long| Democrat)

  $\begin{array}{l}=\frac{169}{360}\\ =0.4694\end{array}$

It is a conditional probability, where, the tables contain $360$ democrats while $169$ of the $360$ democrats are expected to live about as long.

Now, for the other probability, we have,

P (About as long)

  $\begin{array}{l}=\frac{416}{998}\\ =0.4168\end{array}$

Where, the table contains $998$ U.S. adults in total while $416$ of the $998$ U.S. adults are expected to live about as long.

Thus, we noted that from all the above five probabilities the second probability i.e.

P (Independent or not so long)

  $\begin{array}{l}=\frac{470}{998}\\ =0.4709\end{array}$

This is the largest probability of all the other probabilities. Thus, option (b) is the correct option.