Curriculum Core Rules & Strategy
Always write out given values and the relevant formula before substituting numbers.
Substitute your final answer back into the original problem to verify correctness.
Double-check negative signs, common denominators, and arithmetic order of operations.
Mastering Probability Basics with SolveCalc
Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). P(event) = favourable outcomes ÷ total equally likely outcomes. This single formula underlies all of classical probability theory.
The Probability Scale and Classical Definition
Probability is a number between 0 and 1 inclusive that measures the likelihood of an event:
- P = 0: Impossible (rolling a 7 on a standard die)
- P = 0.5: Equally likely to happen or not (fair coin flip)
- P = 1: Certain (rolling a number between 1 and 6)
Classical (theoretical) probability: P(A) = Number of favourable outcomes / Total number of equally likely outcomes.
Complement Rule and Combined Events
Complement rule: P(A does NOT happen) = 1 − P(A). If P(rain) = 0.3, then P(no rain) = 0.7.
Addition rule (OR):
- Mutually exclusive events (can't both happen): P(A or B) = P(A) + P(B)
- Non-mutually exclusive: P(A or B) = P(A) + P(B) − P(A and B)
Multiplication rule (AND):
- Independent events: P(A and B) = P(A) × P(B)
- Dependent events: P(A and B) = P(A) × P(B|A) where P(B|A) is conditional probability
Listing Sample Spaces Systematically
A sample space is the complete set of all possible outcomes. Listing it systematically prevents missed cases:
P(exactly one head) = 2/4 = 1/2
For larger sample spaces, use a probability tree (branch for each outcome at each stage) or a two-way table (for two simultaneous events). These systematic tools guarantee no outcome is missed.
Probability Problems — Three Solved
P(red) = 5/10 = 1/2. P(not red) = 1 − 1/2 = 1/2
Favourable pairs: (1,6)(2,5)(3,4)(4,3)(5,2)(6,1) = 6 pairs. Total = 36. P = 6/36 = 1/6
P(1st red) = 4/10. P(2nd red | 1st red) = 3/9. P = 4/10 × 3/9 = 12/90 = 2/15
Probability Errors That Trick Students
- Adding probabilities for AND: P(A AND B) uses multiplication (for independent events), not addition.
- Ignoring replacement: Probability changes after each draw without replacement — failing to adjust for this is a major error.
- The Gambler's Fallacy: After 10 heads in a row, the probability of tails on the next flip is still 0.5. Each flip is independent.
- Not checking P sums to 1: All probabilities for a complete sample space must sum to exactly 1.
Probability Rules — Fast Recall
- "OR = Add (subtract overlap)" — P(A or B) = P(A) + P(B) − P(both). For mutually exclusive, the overlap is 0.
- "AND = Multiply" — P(A and B) = P(A) × P(B) for independent events.
- "Not = 1 minus" — P(not A) = 1 − P(A). Quick complement shortcut for complex events.
Use SolveCalc's Probability Calculator for combinations, permutations, and binomial distributions.
Common Student Pitfalls & Exam Traps
- Skipping steps: Even experienced mathematicians write every step. Skipping creates opportunities for errors.
- Not checking answers: Always substitute your answer back to verify correctness.
- Confusing similar concepts: Small differences in definitions can require completely different approaches.
- Unit errors: Always keep track of units and convert consistently when working with real-world problems.
SolveCalc Insight
DEEP DIVESolveCalc provides free, browser-based math tools for students and educators at every level. Use our interactive calculators alongside this curriculum guide to verify your calculations and explore concepts hands-on — no sign-up, no downloads, no limits.
Explore with SolveCalc's Math Tools Library→Frequently Asked Questions
What is the probability of flipping heads? +
P(heads) = 1/2 = 0.5 = 50%.
What is the difference between theoretical and experimental probability? +
Theoretical = calculated mathematically. Experimental = measured by running actual trials.