Basic probability

Probability and familiarity

Most students find that probabilities are easiest to understand in the form of percentages. For the ACT test, however, you will need to be comfortable seeing probabilities as percentages, fractions, and decimals. The simplest way to see this is that $100\%$ probability is equal to the probability of $\frac{1}{1}$ or $1$. Another easy way to understand this is by saying that $50\%$ is a $\frac{1}{2}$ chance. We’ll spend our time getting comfortable with more difficult examples.

Probability as fractions

The best way to remember how to do probabilities in the form of fractions is as part over whole. $50\%$ can be seen as $50$ out of $100$, or $\frac{50}{100}$. Thus, $50\%$ can be simplified down to $\frac{1}{2}$. Let’s look at one of the most common examples we see on the test:

You have a bag with $3$ white marbles and $2$ black marbles inside. What is the probability of pulling out one black marble from the bag?

These questions are the easiest we can get because they are single probabilities. Let’s explore the multiple probabilities questions using this same skill.

Multiple probabilities questions

Let’s start by trying to understand how to solve these kinds of questions. Imagine the probability of flipping a coin—it either lands on heads or tails. So, the probability of getting heads is $\frac{1}{2}$, or $50\%$. No matter how many coins we flip, there will always be a $50\%$ chance that we get heads. However, the trick to recognizing these questions is when the problem states that you flip two coins in a row. So now, you are flipping two coins at the same time.

If you flip two coins, should the probability of getting all heads be higher or lower than flipping just one coin?

This is what you need to think of when trying to figure out how to solve these questions. It can be easy to get to the point where you say “I have two coin flips, and each one gives me a $\frac{1}{2}$ probability.” But once you get there, what do you do with the two probabilities?

Read closely: always multiply these probabilities when looking for an overall probability. Many people make the mistake of adding these probabilities, which is wrong. Avoid this mistake! We can see why this is wrong if we try our example: if we want the probability of getting heads both times out of $2$ coin tosses, and we add $\frac{1}{2}+\frac{1}{2}$, then we end up with $1$ ($100\%$).

Always multiply these probabilities: $\frac{1}{2}\ast \frac{1}{2}=\frac{1}{4}$. This makes more sense because the probability should get smaller with the more coins we flip. After all, multiplying a fraction by a fraction is how we reduce fractions in general.

Dynamic probability questions

These are the types of questions that you need to look out for in probabilities because they are the only ones that are a bit different. These questions are the ones where the probabilities change the further you go into the question. It makes much more sense in an example:

You have a bag with $3$ white marbles and $2$ black marbles inside. What is the probability of pulling out one black marble and then one white marble from the bag without putting them back?

Normally we would say that the chance of getting a black marble is $\frac{2}{5}$ and a white marble is $\frac{3}{5}$. However, in this example, we are not putting the first marble back. So, once we take out that first black marble it is gone from the bag, and there will now only be $3$ white marbles and $1$ black marble. The probability of pulling the black marble will still be $\frac{2}{5}$, but the new probability for the white marble will use $4$ as the total marbles in the bag since one black one is gone, and will be $\frac{3}{4}$.

This is the only part of the problem that is changing. Now we can continue to follow the strategy we mentioned earlier, by multiplying the two probabilities together to get the overall probability of pulling these two in a row. What would that overall probability be?

Always remember to multiply the probabilities, not add. If there are multiple probabilities, multiply them!