Unit 14
Statistics and Probability
Class 12 Mathematics
Probability
Class 12 Mathematics – Probability Notes PDF
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NEB / CDC Focus
The Grade 12 focus of this unit is solving probability problems using combinations and solving problems involving conditional probability. The basic event notation, multiplication rule, and independence tests below are included because they are required to solve those problems correctly.
1. Probability Basics
An experiment whose exact outcome cannot be predicted with certainty in advance is called a random experiment.
The set of all possible outcomes is the sample space S. An event A is a subset of S.
For equally likely finite outcomes,
1.1 Addition Rule
If A and B are mutually exclusive, then P(A∩B)=0, so
2. Probability Using Combinations
When selection order does not matter, use combinations to count favorable and total selections.
If all groups of size r are equally likely,
A committee of 3 is selected from 5 boys and 4 girls. Find the probability that exactly 2 girls are selected.
Total committees: ⁹C₃=84.
Favorable: ⁴C₂·⁵C₁=6·5=30.
P=30/84=5/14.
A committee, hand of cards, chosen group, or sample usually ignores order, so combinations are appropriate. If positions or order matter, a permutation model may be needed.
3. Conditional Probability
The probability of event A given that B has already occurred is denoted P(A|B).
Likewise,
A card is drawn from a standard deck. Given that it is a face card, find the probability that it is a king.
There are 12 face cards and 4 kings, all kings being face cards.
P(King | Face)=4/12=1/3.
4. Multiplication Rule of Probability
Rearranging the conditional-probability formula gives
This formula is particularly important in sequential experiments.
A bag has 3 red and 2 blue balls. Two balls are drawn without replacement. Find the probability both are red.
P(R₁)=3/5, P(R₂|R₁)=2/4.
P(R₁∩R₂)=(3/5)(2/4)=3/10.
5. Independent and Dependent Events
Events A and B are independent when occurrence of one does not change the probability of the other.
Events are dependent when occurrence of one changes the probability of the other. Drawing without replacement is a common example.
| Concept | Can both occur? | Main probability condition |
|---|---|---|
| Independent | Yes, usually | P(A∩B)=P(A)P(B) |
| Mutually exclusive | No | P(A∩B)=0 |
Mutually exclusive events are not generally independent. If two non-zero-probability events cannot occur together, knowing one occurred makes the other impossible.
6. Without-Replacement Problems
Without replacement, the total number of objects and category counts change after each draw. Therefore later probabilities are conditional on earlier outcomes.
From 5 red and 3 white balls, two are drawn without replacement. Find the probability of one red followed by one white.
P(R then W)=(5/8)(3/7)=15/56.
If the question asks for one red and one white in any order, include both orders:
7. Worked Examples
Five cards are chosen from 8 red and 4 black cards. Find the probability of exactly 3 red cards.
P=[⁸C₃·⁴C₂]/¹²C₅=(56·6)/792=14/33.
If P(A∩B)=0.18 and P(B)=0.30, find P(A|B).
P(A|B)=0.18/0.30=0.6.
If P(A)=0.5 and P(B|A)=0.4, then
P(A∩B)=0.5×0.4=0.2.
If P(A)=0.6, P(B)=0.5, and A,B are independent:
P(A∩B)=0.3.
If two fair coins are tossed, probability of at least one head is
1−P(no head)=1−1/4=3/4.
8. Problem-Solving Strategy
- Define the event clearly.
- Check whether outcomes are equally likely.
- For unordered selections, count using combinations.
- If the problem says “given that,” identify the conditioning event.
- For sequential events, use the multiplication rule and update counts after each draw.
- Check whether replacement makes events independent or dependent.
- For “at least one,” consider using the complement.
- Ensure the final probability lies between 0 and 1.
9. Common Mistakes
In P(A|B), B becomes the effective sample space.
Use combinations only when order truly does not matter.
Without replacement, probabilities change after each draw.
These are different ideas and use different conditions.
For “one of each” in sequence, include all allowed orders unless the order is specified.
“At least one” is often easiest as 1−P(none).
10. Formula Sheet
11. Important Exam Questions
Short
- Define conditional probability.
- Distinguish independent and mutually exclusive events.
- State the multiplication rule.
- State the combination formula.
Numerical
- Solve probability of committee/card selection using combinations.
- Find conditional probability from event probabilities.
- Solve without-replacement ball/card problems.
- Test whether two events are independent.
- Solve “at least one” problems using complements.
12. One-Minute Revision
- Probability always lies between 0 and 1.
- Use nCr for unordered selections.
- P(A|B)=P(A∩B)/P(B).
- P(A∩B)=P(A)P(B|A).
- Independent events satisfy P(A∩B)=P(A)P(B).
- Without replacement usually creates dependence.
- Mutually exclusive does not mean independent.
- For “at least one,” try the complement.
13. Diagram Practice
- Venn diagram of union/intersection.
- Combination-probability flow.
- Conditional sample-space diagram.
- Sequential probability tree.
- With/without-replacement comparison.
- Probability method checklist.
Also Visit
Original Nepal eNotes page: Probability Notes.
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