“Given means divide”
Conditional probability
P(A|B) divides by P(B), because B has already happened and the sample space has shrunk to B alone.
Mathematics, Class 11 and 12 · Gujarat Board, CBSE, GUJCET and JEE
Probability formulas
Probability rewards reading the question carefully more than any other chapter. Conditional probability, Bayes' theorem and the binomial distribution are the three results asked, and each has one standard shape of problem.
14 formulas, 4 memory hooks, and the 5 mistakes that cost the most marks.
Basic rules
| What | Formula | Symbols and notes |
|---|---|---|
| Probability | P(A) = favourable outcomes / total outcomes | between 0 and 1 |
| Complement | P(A') = 1 − P(A) | the at least one route |
| Addition rule | P(A ∪ B) = P(A) + P(B) − P(A ∩ B) | subtract the overlap once |
| Mutually exclusive | P(A ∩ B) = 0 | so the addition rule loses its last term |
| Independent events | P(A ∩ B) = P(A)P(B) | independence is about influence, not overlap |
Conditional probability
| What | Formula | Symbols and notes |
|---|---|---|
| Conditional probability | P(A|B) = P(A ∩ B)/P(B) | given that B has happened |
| Multiplication rule | P(A ∩ B) = P(B)P(A|B) | rearranged from the line above |
| Total probability | P(A) = ΣP(Ei)P(A|Ei) | over a partition of the sample space |
| Bayes' theorem | P(Ei|A) = P(Ei)P(A|Ei)/ΣP(Ej)P(A|Ej) | working backwards from an effect to a cause |
Distributions
| What | Formula | Symbols and notes |
|---|---|---|
| Random variable | a numerical outcome of an experiment | its distribution lists the probabilities |
| Mean | E(X) = Σxipi | also called the expectation |
| Variance | Var(X) = Σxi²pi − [E(X)]² | and the standard deviation is its square root |
| Binomial distribution | P(X = r) = nCr pr qn−r | n independent trials, q = 1 − p |
| Binomial mean and variance | mean = np, variance = npq | so the variance is never larger than the mean |
Taught by Darshit Bhatt, founder and lead faculty at Darshit Sir Institute. Instrumentation and Control engineer, with a master's degree and a chartered degree, teaching in Paldi, Ahmedabad since 2006.
Know the formula but not when to use it?
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Memory hooks
How to remember this chapter.
“Bayes runs backwards”
When to use Bayes' theorem
If the question tells you the outcome and asks which box or machine it probably came from, it is Bayes. Forwards is total probability; backwards is Bayes.
“At least one means one minus none”
The fastest route in most questions
Counting every successful case is slow. Find the probability of none and subtract from 1.
“np and npq”
The binomial results
Mean np, variance npq. Since q is at most 1, the variance can never exceed the mean, which is a quick sanity check.
Where marks go
The mistakes that cost the most.
- Treating mutually exclusive and independent as the same thing. Mutually exclusive events with non-zero probability are never independent.
- Forgetting to subtract P(A ∩ B) in the addition rule, which double counts the overlap.
- Using Bayes' theorem in the forward direction, where total probability is what is needed.
- Writing the variance as Σx²p without subtracting the square of the mean.
- Using the binomial distribution where the trials are not independent, such as drawing cards without replacement.
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