Probability on a pie chart is just a variant of geometric probability. The probability of an event represented by any sector on the pie chart is equal to the area of that sector divided by the total area of the pie chart. The total probability of all the events represented by the pie chart is always $1$. If there are '$n$' events, then each of the events is represented by a sector on the pie chart. The area of the sector is proportional to the probability of the event. The area of the sector in turn is proportional to the angle subtended by the sector at the centre. 

A sample of $20$ from a population of consisting of students at a school is studied to find out what genre of movies do they like. The results are as shown in the pie chart below:

Probability on a Pie Chart

Now here note that $4$ out of the $20$ students like comedy. Thus the probability that a randomly selected student of that school likes comedy would be:

$\frac{4}{20}$ = $\frac{1}{5}$

In terms of percentage it would be:

$\frac{1}{5}$ $\times\ 100$ = $20 \%$

Thus there is a $20 \%$ chance that a student selected at random from the school likes comedy. Similarly, the probability for an action film being liked is:

$\frac{5}{20}$ = $\frac{1}{4}$ = $25 \%$

For making the pie chart we use the table method that we would have learned in basic statistics. If you don’t remember, don’t worry! We shall be discussing that using an example below.

Word Problems

Example 1: 

At a huge software firm office, a survey was conducted using a random sample of $50$ employees to find out which are the common browsers used by them. The data collected was as follows:

Browser Number of employees using it
Firefox 19
Chrome 18
Opera 8
Safari 2
Internet Explorer 3

Construct a pie chart and answer the following questions:

a) What is the probability that a person selected at random uses opera browser?

b) What percent of the population uses Firefox or Chrome?

Solution:

First we need to construct the pie chart. For that we need the central angle for each of those categories. The central angle would be proportional to the number of employees using that particular browser.

Browser $f_i$ $\theta$
Firefox $19$ $\frac{19}{50}$ $\times\ 360$ = $136.8^{\circ}$
Chrome $18$ $\frac{18}{50}$ $\times\ 360$ = $129.6^{\circ}$
Opera $8$ $\frac{8}{50}$ $\times\ 360$ = $57.6^{\circ}$
Safari $2$ $\frac{2}{50}$ $\times\ 360$ = $14.4^{\circ}$
Internet Explorer $3$ $\frac{3}{50}$ $\times\ 360$ = $21.6^{\circ}$
Total $50$ $360^{\circ}$

Thus using the above $\theta$ values, we shall sketch the pie chart. The central angle for each sector would be the respective θ value. Our pie chart would look as follows:

Probability on a Pie Chart Word Problem

Now let us find the corresponding probability for each of those categories.


Browser $f_i$ Probability
Firefox $19$ $\frac{19}{50}$ = $0.38$
Chrome $18$ $\frac{18}{50}$ = $0.36$
Opera $8$ $\frac{8}{50}$ = $0.16$
Safari $2$ $\frac{2}{50}$ = $0.04$
Internet Explorer $3$ $\frac{3}{50}$ = $0.06$
Total $50$ $1$

Let us now put these probabilities on the pie chart.

Probability on a Pie Chart Word Problems

That completes our pie chart. Now based on this chart, let us answer the questions given.

a) From the chart we see that the probability of opera browser usage is $0.16$ or $16 \%$.

b) From the chart we see that proportion of firefox users is $0.38$ and that of chrome users is $0.36$. So the total proportion of both these two users put together would be:

$0.38 + 0.36$ = $0.74$

Converting that to percentage we have:

$0.74 \times 100$ = $74 \%\ \leftarrow\ Answer!$
Example 2:

A survey was carried out to study the daily power supply of a metro city. The results were as tabulated below:

Power Source Amount of Power (mWatt)
Thermal $3$
Hydro $4$
Wind $3.5$
Solar $5$
Tidal $2.5$
Nuclear $1.5$
Natural gas $0.5$

Construct a pie chart and then answer the following questions.

a) Which energy source has the highest probability of being used?

b) What is the probability that a factory picked at random uses thermal or natural gas as its source of power production?

Solution:

First let us make the pie chart. For that let us find the central angles for each of the categories. For that we shall add a column to the given table.

Power source Amount of power (mWatt) Central angle
Thermal $3$ = $\frac{3}{20}$ $\times\ 360$ = $54^{\circ}$
Hydro $4$ = $\frac{4}{20}$ $\times\ 360$ = $72^{\circ}$
Wind $3.5$ = $\frac{3.5}{20}$ $\times\ 360$ = $63^{\circ}$
Solar $5$ = $\frac{5}{20}$ $\times\ 360$ = $90^{\circ}$
Tidal $2.5$ = $\frac{2.5}{20}$ $\times\ 360$ = $45^{\circ}$
Nuclear $1.5$ = $\frac{1.5}{20}$ $\times\ 360$ = $27^{\circ}$
Natural Gas $0.5$ = $\frac{0.5}{20}$ $\times\ 360$ = $9^{\circ}$
Total $20$ = $360^{\circ}$

Now that we have all the angle measures, we can construct the pie chart.

Probability on a Pie Chart Example
 
Now let us find the probability of each of the categories. For that we divide each of the respective mWatts with the total mWatts which is $20$.


Power Source Amount of power (mWatt) Probability
Thermal $3$ = $\frac{3}{20}$ = $0.15$
Hydro $4$ = $\frac{4}{20}$ = $0.2$
Wind $3.5$ = $\frac{3.5}{20}$ = $0.175$
Solar $5 $ = $\frac{5}{20}$ = $0.25$
Tidal $2.5$ = $\frac{2.5}{20}$ = $0.125$
Nuclear $1.5$ = $\frac{1.5}{20}$ = $0.075$
Natural Gas $0.5$ = $\frac{0.5}{20}$ = $0.025$

Now let us mark the probabilities on the pie chart.

Probability on a Pie Chart Examples
 
Now let us try to answer the questions given.

Since the highest probability is 0.25 which belongs to solar. Thus, solar energy has the highest probability of being used.

The probability of thermal power is 0.15 and that of natural gas is 0.025. Thus the combined probability that a factor picked at random uses thermal or natural gas would be:

$0.15 + 0.025$

= $0.175\ \leftarrow\ Answer!$
Example 3:

The following pie chart shows the distribution of car sales amongst $6$ six companies.

Probability on a Pie Chart Solved Example
 
Answer the following questions based on the above pie chart.

a) If a car is selected at random, what is the probability that it is either a Maruti or a GM variant?

b) If the total number of cars in a town is $150,000$, then how many of them can be expected to be Hyundai variants?

c) Jimmy’s car is very old. The probability of any other person owing the same car is $0.12$. Which car is this?

Solution:

a) The probability of a Maruti is $24 \%$ and that of a GM is $35 \%$. So the total combined probability of both these would be = $24 + 35$ = $59 \%$

b) The probability of Hyundai is $15 \%$. So out of $150,000$ cars, the possible number of Hyundai cars would be:

$150,000\ \times\ 15 \%$

= $150,000\ \times$ $\frac{15}{100}$

= $22,500$ cars

c) The probability of Jimmy’s cars is $0.12$. That is same as $12 \%$. The only car that has that probability is Fiat. So Jimmy’s car has to be a Fiat variant.