Math & History: Independent Events and the Multiplication Rule

I. Intro to Independent & Dependent Events

Independent Events
In statistics, the term independent event means to have one event not dependent on the other. In other words, two events A and B, are independent if the occurrence of one does not affect the probability of the other.

Dependent Events

Events A and B are dependent if the occurrence of Event A changes the probability of Event B.

Example

Imagine you have a standard deck of 52 cards. You want to draw a King (Event A), and then draw another King (Event B)

1. Independent (With Replacement)

Action: You pick a card, look at it, and put it back in the deck before picking the next one.
Event A: You draw a King. Probability is 4 in 52.
The Reset: You put the King back and shuffle. The deck is back to 100% normal.
Event B: You draw again. The probability of getting a King is still 4 in 52.
Result: The first event didn't change the deck, so the probability stayed the same. Independent.

2. Dependent (Without Replacement)

Action: You pick a card, look at it, and keep it in your pocket.
Event A: You draw a King. Probability is 4 in 52.
The Change: You keep the card. Now there are only 51 cards left in the deck, and only 3 Kings left.
Event B: You draw again. The probability is now 3 in 51.
Result: Because you didn't put the first card back, the odds for the second draw changed. Dependent.

II. Independent Events

A. Using the Multiplication Rule for Independent Events

In ABC High School, 30 percent of the students have a part-time job, and 25 percent of the students from the high school are on the honor roll. Event A represents randomly choosing a student holding a part-time job. Event B represents randomly choosing a student on the honor roll. What is the probability of both events occurring?

Event A is randomly choosing a student holding a part-time job, and event B is randomly choosing a student on the honor roll. These 2 events are independent of each other. In other words, whether you hold a part-time job is not dependent on your being on the honor roll, or vice versa. The outcome of one event is not dependent on the outcome of the second event. To calculate the probability, you would look at the overlapping part of the Venn diagram. The region representing A and B is the probability of both events occurring. Let’s look at the probability calculation, which is done with the Multiplication Rule:

In other words, under independence, the probability of randomly selecting a student who both has a part-time job and is on the honor roll is 7.5%.

Multiplication Rule For Independent Events

If A and B are independent events, the probability of both events occurring is the product of the probabilities of the individual events.

P(A∩B) = P(A and B) = P(A) • P(B)

Example 1

2 coins are tossed one after the other. Event A consists of the outcomes when tossing heads on the first toss. Event B consists of the outcomes when tossing heads on the second toss. What is the probability of both events occurring?

Event A consists of the outcomes when getting heads on the first toss, and event B consists of the outcomes when getting heads on the second toss. What would be the probability of tossing the coins and getting a head on both the first coin and the second coin? We know that the probability of getting a head on a coin toss is 1/2, or 50%. In other words, we have a 50% chance of getting a head on a toss of a fair coin and a 50% chance of getting a tail.

Therefore, there is a 25% chance of getting 2 heads when tossing 2 fair coins.

Example 2

The following table represents data collected from a grade 12 class in DEF High School.

Suppose 1 student was chosen at random from the grade 12 class.

(a) What is the probability that the student is female?

(b) What is the probability that the student is going to university?

Now suppose 2 people both randomly chose 1 student from the grade 12 class. Assume that it's possible for them to choose the same student.

(c) What is the probability that the first person chooses a student who is female and the second person chooses a student who is going to university?

Answers

(a) P(female) = 80/164 = 0.4878 or 48.78% chance the student is female

(b) P(going to university) = 71/164 = 0.4329 or 43.29% chance the student is going to university

48.78% chance female x 43.29% chance going to university = 21.11%

Therefore there is a 21.11% probability that the first person chooses a student who is female and the second person chooses a student who is going to university.

Example 3

2 cards are chosen from a deck of cards. The first card is replaced before choosing the second card. What is the probability that they both will be sevens?

Let A = 1st seven chosen

Let B = 2nd seven chosen

A deck of cards consists of 52 cards. Each deck has 4 suits (with each suit having13 cards). Each suit has one seven, therefore there are 4 sevens in each deck of 52 cards.

Since the card is replaced after the first selection, these events are independent.

P(A) = 4/52

P(B) = 4/52

P(A and B) = 4/52 x 4/52 or P(A∩B) = 4/52 x 4/52

P(A∩B) = 16/2704 = 1/169

B. Conceptual Understanding

One way to help conceptually understand the multiplication rule is with the use of a tree diagram.

Let's go back to example 1. What is the probability of flipping a fair coin and getting "heads" twice in a row? That is, what is the probability of getting heads on the first flip AND heads on the second flip?

Imagine we had100 people simulate this and flip a coin twice. On average, 50 people would get heads on the first flip, and then 25 of them would get heads again. So 25 out of the original 100 people — or 1/4 of them — would get heads twice in a row.

The number of people we start with doesn't really matter. Theoretically, 1/2 of the original group will get heads, and 1/2 of that group will get heads again. To find a fraction of a fraction, we multiply.

We can represent this concept with a tree diagram like the one shown below.