Finding the Probability of Two Events

Sometimes, the occurrence of one event affects the occurrence of another. If this is the case, the events are said to be dependent. If not, they are independent.

Two events are said to be dependent when the occurrence of one affects the occurrence of the other. For example, consider drawing two marbles from a bowl, one at a time.

The probability of first picking a green marble can be calculated by dividing the favorable outcomes by the possible outcomes. There is $1$ green marble and $3$ total marbles. $$P(\text{green})=\frac{1}{3}$$ Suppose that the first marble is replaced before the second draw. Therefore, after the replacement, there is $1$ purple marble and $3$ total marbles. $$P(\text{purple})=\frac{1}{3}$$ The combined probability of picking a green marble first and a purple marble second can be calculated using the Multiplication Rule of Probability. $$P(\text{green then purple})=\frac{1}{3}\cdot \frac{1}{3}=\frac{1}{9}$$ These events are not dependent. Suppose instead that, after the green marble is picked, it is not replaced in the bowl.

This affects the probability of picking a purple marble on the second draw. Now, there still is $1$ purple marble but, instead of $3,$ there are $2$ total marbles. $$P(\text{purple})=\frac{1}{2}$$ With this information, the probability of picking green and then purple can be calculated. $$P(\text{green then purple})=\frac{1}{3}\cdot \frac{1}{2}=\frac{1}{6}$$

As can be seen, these events are dependent because the occurrence of the first affects the occurrence of the second.

When categorical data belongs to two categories, such as if people are asked whether they own a car and whether they have a driver's license, it can be presented in a two-way frequency table. One of the categories is represented by the rows of the table, and the other by the columns. The above survey, with $100$ participants, could result in the following answers.

		Driver's license
		Yes	No
Car	Yes	$43$	$4$
	No	$24$	$29$

The two categories are then "car" and "driver's license," both with the possible answers "yes" and "no." The entries in the table are called joint frequencies. Often, two-way frequency tables include the total of the rows and columns. These totals are called marginal frequencies. The sum of the "total" row and "total" column are each equal to the sum of all joint frequencies, $100$ in this case.

		Driver's license
		Yes	No	Total
Car	Yes	$43$	$4$	$47$
	No	$24$	$29$	$53$
	Total	$67$	$33$	$100$

From the table, it can, for instance, be read that $43$ out of the $100$ people both own a car and have a driver's license, and that $33$ of the $100$ do not have a driver's license.

A joint relative frequency is the ratio of a joint frequency and the total number of values or observations. Similarly, a marginal relative frequency is the ratio of a marginal frequency and the total. For the example above, the joint and marginal relative frequencies are found by dividing the frequencies by $53,$ the number of participants.

		Hat preference
		Top hat	Beret	Total
Gender	Male	$\frac{6}{53}\approx 0.11$	$\frac{12}{53}\approx 0.23$	$\frac{18}{53}\approx 0.34$
	Female	$\frac{15}{53}\approx 0.28$	$\frac{20}{53}\approx 0.38$	$\frac{35}{53}\approx 0.66$
	Total	$\frac{21}{53}\approx 0.40$	$\frac{32}{53}\approx 0.60$	$1$

Notice that, ignoring the error margin introduced by rounding, a marginal relative frequency can be found by adding a row or column of joint marginal frequencies. This table shows, for instance, that female's with a preference for berets make up about $38\%$ of the participants.

A conditional relative frequency is the ratio of a joint frequency and either of its corresponding two marginal frequencies. Alternatively, it can be calculated using relative joint and marginal frequencies. As an example, the following data will be used.

		Driver's license
		Yes	No	Total
Car	Yes	$43$	$4$	$47$
	No	$24$	$29$	$53$
	Total	$67$	$33$	$100$

Using the column totals, the left column of joint frequencies should be divided by $67,$ and the right column by $33.$ Since the column totals are used, the sum of the conditional relative frequencies will be $1.$

		Driver's license
		Yes	No
Car	Yes	$\frac{43}{67}\approx 0.64$	$\frac{4}{33}\approx 0.12$
	No	$\frac{24}{67}\approx 0.36$	$\frac{29}{33}\approx 0.88$

This table shows that, for instance, out of all the participants with a driver's license, about $64\%$ of them own a car, and out of those without a driver's license, $88\%$ do not have a car.