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Probability Word Problems
This lesson shows how to read a probability word problem, identify the total outcomes and the favorable outcomes, and write the probability as a fraction. Examples cover coins, dice, marbles, and spinners so you can apply the same method to any story problem about chance.
The Probability Formula
For any event, probability is written as a fraction between \( 0 \) and \( 1 \):
\( P = \dfrac{f}{n} \)
Here \( f \) stands for the number of favorable outcomes (the outcomes that match what the question is asking about) and \( n \) stands for the total number of possible outcomes. A probability of \( 0 \) means the event is impossible, and a probability of \( 1 \) means the event is certain. Everything else falls somewhere in between, and word problems often ask you to compare how close a probability is to \( 0 \), \( \dfrac{1}{2} \), or \( 1 \).
Sometimes it is easier to find the probability that something does not happen. The complement rule says:
\( P(A') = 1 - P(A) \)
This is useful whenever counting the outcomes you do not want is faster than counting the outcomes you do want.
A Step-by-Step Method for Any Word Problem
Use the same four steps every time you meet a new probability story:
1. Read the problem and list every possible outcome, so you know the value of \( n \).
2. Underline or count the outcomes that match the event described, giving you \( f \).
3. Write the fraction \( \dfrac{f}{n} \) and simplify it to lowest terms.
4. Interpret the answer in the words of the problem, for example as a fraction, a decimal, or a percent.
Before you can even reach step one, you sometimes need to organize the information given in the problem. Skills like collecting data help you sort a messy list of results into the outcome counts you need for step one and step two.
Worked Example 1: Marbles in a Bag
A bag contains 3 red marbles, 2 blue marbles, and 5 green marbles. What is the probability of picking a red marble?
Total outcomes: \( n = 3 + 2 + 5 = 10 \).
Favorable outcomes (red marbles): \( f = 3 \).
\( P_r = \dfrac{3}{10} \)
The probability of drawing a red marble is \( \dfrac{3}{10} \), which can also be written as \( 0.3 \) or \( 30\% \).
Worked Example 2: Spinner Problem
The spinner below is divided into four equal sections: two yellow, one blue, and one green. What is the probability of landing on yellow?
Total outcomes: \( n = 4 \) equal sections.
Favorable outcomes (yellow): \( f = 2 \).
\( P_y = \dfrac{2}{4} = \dfrac{1}{2} \)
There is a \( \dfrac{1}{2} \) chance, or \( 50\% \), that the spinner lands on yellow. Notice that the probability of not landing on yellow can be found with the complement rule: \( 1 - \dfrac{1}{2} = \dfrac{1}{2} \), which also equals the chance of landing on blue or green combined.
Common Outcome Sets to Know
Many word problems reuse the same familiar objects, so it helps to memorize their outcome counts:
A coin toss has 2 outcomes: heads or tails.
A standard die has 6 outcomes: the numbers 1 through 6.
A standard deck of playing cards has 52 outcomes, split into 4 suits of 13 cards each.
A spinner or bag of counters has as many outcomes as the number of equal sections or objects described in the problem.
Word problems that talk about results being "likely," "unlikely," "certain," or "equally likely" are using probability vocabulary directly; if you need a refresher on comparing those descriptions, see the lesson on comparative language.
Worked Example 3: Comparing Two Events
A number cube labeled 1 to 6 is rolled once. Which is more likely: rolling an even number, or rolling a number greater than 4?
Even numbers on the cube: 2, 4, 6, so \( f = 3 \) out of \( n = 6 \), giving \( P = \dfrac{3}{6} = \dfrac{1}{2} \).
Numbers greater than 4: 5, 6, so \( f = 2 \) out of \( n = 6 \), giving \( P = \dfrac{2}{6} = \dfrac{1}{3} \).
Since \( \dfrac{1}{2} \) is greater than \( \dfrac{1}{3} \), rolling an even number is more likely than rolling a number greater than 4.
Tips for Reading Probability Word Problems
Watch for the word "or," which usually means you add the favorable outcomes together, and the word "not," which usually signals the complement rule. Draw a quick picture or list, such as the spinner or bag above, whenever the problem describes several categories at once; this is the same organizing habit used in lessons on bar graphs, where sorting data into categories makes the totals easy to see. Finally, always double-check that your fraction is between 0 and 1, since a probability outside that range means an outcome was miscounted.