Learn probability word problems
Probability describes how likely an event is, from impossible through certain. For equally likely outcomes, probability is favorable outcomes divided by total outcomes.
You will turn everyday probability language into favorable outcomes over total outcomes.
The idea in plain English
Probability measures how likely an event is. Zero means impossible; one means certain.
For equally likely outcomes, count the results you want and divide by all possible results.
How to recognise this kind of question
- The question uses words such as chance, likely, randomly, or probability.
- A set of possible outcomes is described.
- You must identify which outcomes count as success.
Quick decision: Write what you want on top and every possible equally likely outcome on the bottom.
The method
- Identify the event and all equally likely possible outcomes.
- Count the outcomes that satisfy the event.
- Write favorable outcomes over total outcomes and reduce the fraction.
- Check whether the result matches the event's likelihood description.
Worked examples
A bag contains 3 red and 9 blue counters. Find P(red).
- There are 3 favorable outcomes and 12 total.
Answer:
A bag has 3 red and 5 blue counters. Find P(red).
- Favorable red counters: 3.
- Total counters: 3 + 5 = 8.
Answer:
Your turn—with help
A fair die is rolled. What is P(number greater than 4)?
Need a hint?
List the die results greater than 4.
Show the answer and steps
- Favorable outcomes are 5 and 6: 2 outcomes.
- There are 6 total outcomes.
Answer:
Watch out for this mistake
Incorrect:
Why it fails: The denominator must include every possible counter, not only the unwanted ones.
Do this instead: There are 6 total counters, so P(red) = 2/6 = 1/3.
Quick check
Try these without looking. Then open each answer to check your thinking.
P(heads) on one fair coin toss
Need a hint?
One favorable side out of two.
Show the answer and steps
- Heads is 1 outcome; total is 2.
Answer:
P(rolling a 3 on a fair die)
Need a hint?
Only one face shows 3.
Show the answer and steps
- 1 favorable outcome over 6 total.
Answer:
Bag: 2 green, 3 yellow. P(green)
Need a hint?
Total the counters first.
Show the answer and steps
- 2 green out of 5 total.
Answer:
Common mistakes
- Using unfavorable outcomes as the numerator.
- Forgetting to include every outcome in the total.
- Calling a probability greater than one-half unlikely.
Practice tips
- State the event in words before counting.
- Estimate whether the answer should be below, equal to, or above one-half.