Probability: Independent Events

Probability tells us the likelihood of an outcome, or certain outcomes. Consider, for example, tossing a coin.

When one tosses a coin once, there are two outcomes: heads or tails. The following probability tree shows us the possible outcomes if a coin is tossed twice in a row.

Let’s say we wanted to find the probability of getting heads both times. We can see that the probability of getting this outcome (HH), is 1/4 or 25% because we see this result only once out of the four possible outcomes (HH, HT, TH, TT).

We can also say that the probability of getting heads and tails after two tosses is 2/4 (simplifies to 1/2) or 50%, because there are two outcomes that satisfy this description (HT, TH).

If you were asked to predict the number of times heads would be the result for 10 coin tosses, what would you say? Most people would say 5/10, because that is the theoretical probability. The theoretical probability is an expectation. However, the theoretical probability does not always match the experimental probability.

In real life, if you were to do a coin toss 10 times, you could have 6 heads and 4 tails, or you could have 3 heads and 7 tails. The experimental probability is the probability found by repeating an experiment (coin tosses) and recording the outcomes.

Let’s consider a slightly more complicated situation:

Imagine you have two bags with their contents described below. At random, you will select one marble from the first bag and then one marble from the second bag. What is the probability that you will select a blue marble from the first bag and a blue marble from the second bag?

Bag 1: 2 blue marbles, 1 red marble

Bag 2: 2 blue marbles, 1 red marble, 1 purple marble

The probability of you selecting a blue marble from the first bag is 2/3, because there are 2 blue marbles and 3 marbles in total.

The probability of you selecting a blue marble from the second bag is 2/4 because there are 2 blue marbles and 4 marbles in total.

When we look at the tree diagram made for this problem, we see there are 12 outcomes, and that there are 4 outcomes where a blue marbles is picked out of both the first and second bag. Therefore, there probability of picking 2 blue marbles in a row are 4 in 12 (or simplified to ⅓).

With this problem, we see two key features that we can apply to other probability problems:

(1) To find the number of outcomes, we can multiply the number of choices or outcomes for one action by the number of choices or outcomes for another. This is called the fundamental counting principle. This principle also works for more than 2 actions.

In our example above, there were 3 possible outcomes for the first bag (picking a blue, a blue, or a red marble) and 4 possible outcomes for the second bag (picking a blue, a blue, a red, or a purple marble).

(2) To find the probability of picking two blue marbles in a row, we can multiply the probability of picking a blue marble out of the first bag (2/3) and the probability of picking a blue marble out of the second bag (2/4).

This is an application of the general multiplication rule, as seen below.

We should also take note that in both the coin-toss and marbles examples, our events were independent.

• If I toss a coin two times, the result or outcome of my first toss has no effect on the outcome of my second toss. Whatever the outcome of my first toss, the probability of my getting a heads or tails does not change for the second toss.

No matter the outcome of my first toss (heads or tails) there is still a 50% chance I get a heads on the second toss, and a 50% I get a tails.

• The marble I pick out of the first bag does not affect or have any impact on the marble I pick out of the second bag. As such, the probability of the second event is not impacted by outcome of the first event.

Probability: Dependent Events

Let’s now consider a different example:

Imagine you have one bag with 6 marbles: 2 blue marbles, 2 red, and 2 purple. You select one marble from the bag, put it back, and then select another marble in the bag. What is the probability that you will select a blue marble twice in a row?

For the first pull, the probability of you selecting a blue marble is 2/6 or 1/3. For the second pull, the probability of you selecting a blue marble is still 2/6 because you put the first marble back.

Using the general multiplication rule, the probability of you selecting two blue marbles in a row is found by:

Therefore, the probability of you selecting two marbles in a row is 1/9.

But let’s make a small change to the scenario:

You select one marble from the bag and without replacing the first, you select another. What is the probability that you will select 2 blue marbles from the bag.

For simplicity’s sake, let’s call selecting a blue marble on the first pull Event A. Let’s then call selecting a blue marble on the second pull Event B.

The probability of Event A is still 2/6 or 1/3 because there are 6 marbles and 2 are blue - just like in the previous example. However, *if* you select a blue marble on the first pull (or in other words, Event A occurs), the probability of selecting a blue marble on the second pull (Event B) is now 1/5. Consider the below image to see why this is true.

In this situation, we can still use the general multiplication rule to find the probability that 2 blue marbles are selected in a row, but we must make a slight change to it. In this case, the general multiplication rule is amended as follows:

The second term, P(B|A) means “the probability of Event B, given that Event A has happened”.

Earlier, we defined Event A as selecting a blue marble on the first pull, and Event B as selecting a blue marble on the second pull.

So, to determine the probability of both Event A and B occurring (selecting two blue marbles in a row), we must make us of the following two probabilities:

• P(A) - which is the probability of selecting a blue marble on the first pull

• P(B | A) - the probability of selecting a blue marble, given that a blue marble was selected on the first pull

We saw above that P(B|A) is 1/5, because *if* a blue marble is selected on the first pull, we have 5 marbles left in the bag and only one of them is blue.

Therefore, we can calculate the probability of two blue marbles selected in a row, as:

Therefore, the probability of selecting two blue marbles in a row is 1/15. In this example, the events are considered to be dependent because the probability of Event B is affected by, or changed as a result of, Event A.

• Two events are independent if the result of the second event is not affected by the result of the first event.

The probability of two independent events is given by:

• Two events are dependent if the result of the first event affects the outcome of the second event so that the probability is changed.

The probability of two dependent events is given by:

• Probabilities can be represented with fractions, ratios, decimals, and percentages.