How to Solve Probability Problems

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Probability measures how likely an event is. On ACT problems, first identify the successful outcomes and the total possible outcomes.

Core Rules

For equally likely outcomes, \(P(E)=\frac{\text{favorable outcomes}}{\text{total outcomes}}\).
The complement rule is \(P(\text{not }E)=1-P(E)\).
For independent events, multiply probabilities.
For events without replacement, update the total after each draw.

Worked Example

If a bag has \(5\) red marbles and \(7\) blue marbles, then \(P(\text{red})=5/12\). The probability of not red is \(7/12\).

Video Lesson

Probability Problems

Think of this lesson as more than a rule to memorize. Probability Problems is about organizing data and choosing the right summary measure. A strong student does not rush to the first formula on the page; they pause, identify the structure of the problem, and then choose the tool that matches that structure. That pause is what prevents most mistakes.

Probability compares favorable outcomes to total possible outcomes: \(P(event)=\frac{favorable}{total}\), assuming the outcomes are equally likely.

Here is the teacher way to approach the topic. First, read the problem slowly and underline the information that is actually given. Next, name the target: are you finding a value, simplifying an expression, comparing two quantities, solving for a variable, or interpreting a graph? Once the target is clear, the calculation becomes much less mysterious because every step has a job.

Read what is given and what is being asked.
Choose the rule that connects them.
Substitute carefully and simplify in small steps.
Check the final answer against the original question.

A helpful habit is to say what each number represents before using it. For example, if a number is a denominator, a radius, a slope, a common difference, or a coefficient, it should not be treated like an ordinary loose number. Its role tells you where it belongs in the formula. This is especially important on ACT-style questions because many wrong answer choices come from using the right numbers in the wrong places.

Another good habit is to keep the work organized vertically. Write one clean line for substitution, one line for simplifying, and one line for the final answer. If the problem has units, keep the units attached. If the problem has signs, exponents, or parentheses, copy them carefully from one line to the next. Most errors in this topic are not caused by a hard idea; they are caused by dropping a negative sign, combining unlike terms, using the wrong denominator, or skipping a check.

When you finish, ask a quick reasonableness question. Should the answer be positive or negative? Should it be larger or smaller than the original number? Does it fit the graph, table, shape, or equation? Can you plug it back into the original problem? This final check turns the lesson from a procedure into understanding.

On a test, the goal is not to write the longest solution. The goal is to write enough clear work that you can see the structure, avoid traps, and recover quickly if one line goes wrong. Practice the examples below with that mindset: identify the type of problem, choose the matching rule, show the substitution, simplify carefully, and check the answer before moving on.

Exercises

Solve each ACT-style practice problem. The questions increase in difficulty.

1) A bag has 3 red and 5 blue marbles. What is the probability of drawing a red marble?

2) A fair die is rolled. What is the probability of rolling an even number?

3) A card is chosen from a standard 52-card deck. What is the probability it is a heart?

4) A spinner has 8 equal sections numbered 1 through 8. What is the probability of landing on a number greater than 5?

5) A jar has 6 green, 4 yellow, and 10 black beads. What is the probability of not drawing black?

6) Two fair coins are tossed. What is the probability of getting exactly one head?

7) A fair die is rolled twice. What is the probability of rolling a 6 both times?

8) A bag has 5 red and 7 blue marbles. Two marbles are drawn without replacement. What is the probability both are red?

9) From numbers 1 through 20, one number is chosen. What is the probability it is a multiple of 3 or 5?

10) A class has 12 juniors and 18 seniors. If one student is chosen, what is the probability the student is a senior?

11) A password digit is chosen from 0 through 9. What is the probability it is prime?

12) A fair die is rolled. What is the probability of not rolling a number less than 3?

13) A basket has 4 apples, 5 oranges, and 6 pears. What is the probability of choosing an orange or a pear?

14) Two cards are drawn without replacement from a standard deck. What is the probability both are aces?

15) A fair coin is tossed and a fair die is rolled. What is the probability of tails and an odd number?

16) A box has 8 red, 6 blue, and 6 green tiles. If one tile is chosen, what is the probability it is blue given that it is not red?

17) In a group, 14 students take art, 18 take music, and 6 take both. If 40 students are surveyed, what is the probability a student takes art or music?

18) A bag has 3 red, 4 blue, and 5 white marbles. Three marbles are drawn without replacement. What is the probability all three are white?

19) A committee of 2 is chosen from 5 boys and 4 girls. What is the probability both chosen are girls?

20) A survey shows 30 percent of students play soccer and 20 percent play both soccer and band. If a student who plays soccer is chosen, what is the probability the student also plays band?

Answers

1) There are \(3\) favorable red marbles and \(8\) marbles total.
Therefore the probability is \(\frac{3}{8}\).

2) The even outcomes are \(2,4,6\), so there are \(3\) favorable outcomes out of \(6\).
Therefore the probability is \(\frac{1}{2}\).

3) There are \(13\) hearts in \(52\) cards.
Therefore the probability is \(\frac{1}{4}\).

4) The favorable outcomes are \(6,7,8\), or \(3\) outcomes out of \(8\).
Therefore the probability is \(\frac{3}{8}\).

5) Not black means green or yellow: \(6+4=10\) favorable beads out of \(20\).
Therefore the probability is \(\frac{1}{2}\).

6) The outcomes are HH, HT, TH, TT. Exactly one head occurs in HT and TH.
Therefore the probability is \(\frac{1}{2}\).

7) Each roll has probability \(1/6\), and the rolls are independent.
Therefore the probability is \(\frac{1}{36}\).

8) First red is \(5/12\). Then \(4\) red remain out of \(11\).
Therefore the probability is \(\frac{5}{33}\).

9) Multiples of 3 or 5 are \(3,5,6,9,10,12,15,18,20\), so \(9\) numbers qualify.
Therefore the probability is \(\frac{9}{20}\).

10) There are \(18\) seniors out of \(30\) students.
Therefore the probability is \(\frac{3}{5}\).

11) The prime digits are \(2,3,5,7\), so \(4\) of \(10\) digits qualify.
Therefore the probability is \(\frac{2}{5}\).

12) Less than \(3\) means \(1\) or \(2\). The complement has \(4\) outcomes: \(3,4,5,6\).
Therefore the probability is \(\frac{2}{3}\).

13) Orange or pear gives \(5+6=11\) favorable fruits out of \(15\).
Therefore the probability is \(\frac{11}{15}\).

14) First ace: \(4/52\). Second ace: \(3/51\).
Therefore the probability is \(\frac{1}{221}\).

15) Tails has probability \(1/2\); odd on a die has probability \(3/6\).
Therefore the probability is \(\frac{1}{4}\).

16) Given not red, only \(6+6=12\) blue or green tiles remain possible; \(6\) are blue.
Therefore the probability is \(\frac{1}{2}\).

17) Use inclusion-exclusion: \(14+18-6=26\) students take art or music.
Therefore the probability is \(\frac{13}{20}\).

18) Multiply changing probabilities: \(5/12\), then \(4/11\), then \(3/10\).
Therefore the probability is \(\frac{1}{22}\).

19) There are \(\binom{4}{2}\) girl pairs and \(\binom{9}{2}\) total pairs.
Therefore the probability is \(\frac{1}{6}\).

20) Conditional probability divides the overlap by the condition: \(20\%/30\%\).
Therefore the probability is \(\frac{2}{3}\).

More Practice

Free printable Worksheets

How to Solve Probability Problems