Binomial Probabilities
A free Statistics and Data Analysis lesson from the “Probability and Distributions” unit, with a worked example and practice problems including step-by-step solutions.
A binomial probability combines counting and multiplication: choose which trials are successes, then multiply by the probability of that pattern. This lesson builds the habit of reading the context first, choosing the right statistical tool, calculating carefully, and then writing what the result means. By the end, students should be able to do the computation and explain why that computation answers the question.
What you'll learn
- Compute simple binomial probabilities
- Use combinations for exactly k successes
- Interpret binomial results in context
Worked example
Problem. If P(A) = 0.35, what is P(not A)?
- Complements add to 1.
- Subtract P(A) from 1.
Answer: 0.65
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.
Practice problems
1. Practice case A: If P(A) = 0.35, what is P(not A)?
Show solution
- Complements add to 1.
- Subtract P(A) from 1.
Answer: 0.65
2. Practice case B: Events A and B are disjoint. If P(A) = 0.2 and P(B) = 0.5, what is P(A or B)?
Show solution
- For disjoint events, add the probabilities.
- There is no overlap to subtract.
Answer: 0.7
3. Practice case C: Among 15 students in a group, 5 play soccer. What is P(plays soccer | in this group)?
Show solution
- The condition sets the denominator to this group.
- 5/15 simplifies to 1/3.
Answer: 1/3
4. Practice case D: If two events are independent, then P(B | A) equals:
Choices: P(B) · P(A) + P(B) · 0 · P(A) - P(B)
Show solution
- Independence means knowing A happened does not change B.
- So P(B | A) = P(B).
Answer: P(B)
5. Practice case E: A fair coin is flipped 3 times. What is the probability of HHH?
Show solution
- Each heads has probability 1/2.
- (1/2)^3 = 1/8.
Answer: 1/8
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