Basic probability

Probability and familiarity

Most students find probabilities easiest to understand as percentages. On the ACT, though, you’ll need to be comfortable switching between percentages, fractions, and decimals.

A few key equivalences help you translate between forms:

• $100\%$ probability means something is certain. As a fraction, that’s $\frac{1}{1}$, and as a decimal, it’s $1$.
• $50\%$ probability means something happens half the time. As a fraction, that’s $\frac{1}{2}$.

In this section, you’ll practice moving between these forms and handling more challenging probability setups.

Probability as fractions

A reliable way to write probability as a fraction is part over whole:

• Part = the number of outcomes you want
• Whole = the total number of possible outcomes

For example, $50\%$ means $50$ out of $100$, or $\frac{50}{100}$, which simplifies to $\frac{1}{2}$.

Let’s look at a common ACT-style example:

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 are the simplest probability questions because they involve a single probability. Next, you’ll use the same idea in multiple probabilities questions.

Multiple probabilities questions

To see how these work, start with a coin flip. A coin lands on heads or tails, so the probability of heads is $\frac{1}{2}$ (or $50\%$).

Now suppose you flip two coins in a row (or flip two coins at the same time). Each flip still has a $\frac{1}{2}$ chance of landing on heads, but the question is asking for the probability that both flips come up heads.

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

Here’s the key rule:

Read closely: when you want the probability that multiple events all happen, you multiply their probabilities.

A common mistake is to add. For example, if you add $\frac{1}{2}+\frac{1}{2}$, you get $1$ (or $100\%$), which would incorrectly suggest you’re guaranteed to get two heads.

Instead, multiply:

Always multiply these probabilities: $\frac{1}{2}\ast \frac{1}{2}=\frac{1}{4}$.

This makes sense because the probability should get smaller when you require more things to happen.

Dynamic probability questions

These are the probability questions to watch for because the probabilities change as you go. This happens when the situation changes after the first event - most commonly when something is removed and not replaced.

An example makes this clear:

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?

At the start:

• Probability of black first: $\frac{2}{5}$

But you are not putting the first marble back. If the first marble is black, then one black marble is removed. The bag now has:

• $3$ white marbles
• $1$ black marble
• $4$ total marbles

So the probability of white second (after removing a black) is:

• $\frac{3}{4}$

Now multiply to get the overall probability of both events happening in order:

What would that overall probability be?

Always remember: if there are multiple probabilities and you want them all to happen, multiply, don’t add.