Concept Core
From classical probability to Bayes' theorem β the complete framework.
Classical Probability & Fundamental Rules
Classical definition: P(A) = (favourable outcomes)/(total equally likely outcomes)
0 β€ P(A) β€ 1 P(A) + P(A') = 1
P(A βͺ B) = P(A) + P(B) β P(A β© B)
Mutually exclusive events: A β© B = β
β P(A βͺ B) = P(A) + P(B)
Independent events: P(A β© B) = P(A) Γ P(B)
Conditional Probability
P(A|B) = P(A β© B) / P(B) (B has occurred)
P(A|B) β P(B|A) in general. For independent events: P(A|B) = P(A) β B's occurrence doesn't affect A.
Multiplication rule: P(A β© B) = P(A) Γ P(B|A) = P(B) Γ P(A|B)
Total Probability Theorem
If Bβ, Bβ, ..., Bβ are mutually exclusive and exhaustive events:
P(A) = Ξ£ P(Bα΅’) Γ P(A|Bα΅’)
Used when the sample space is partitioned and we know conditional probabilities in each partition.
Bayes' Theorem
Given event A has occurred, find which partition Bβ caused it:
P(Bβ|A) = P(Bβ) Γ P(A|Bβ) / Ξ£ P(Bα΅’) Γ P(A|Bα΅’)
The numerator is the individual term; denominator is P(A) from total probability theorem.
Binomial Distribution
For n independent trials, each with probability p of success:
P(X = r) = βΏCα΅£ pΚ³ (1βp)βΏβ»Κ³ = βΏCα΅£ pΚ³ qβΏβ»Κ³
Mean = np Variance = npq SD = β(npq)
q = 1 β p. Sum of all P(X=r) = 1. Symmetric when p = q = Β½.
Geometric & Other Distributions
Geometric distribution: P(first success on rth trial) = q^(r-1) Γ p
Random variable expectation: E(X) = Ξ£ xα΅’ P(xα΅’). For functions: E(aX+b) = aE(X) + b
Variance: Var(X) = E(XΒ²) β [E(X)]Β². Always β₯ 0.
Formula Vault
All probability formulas β classical, conditional, Bayes', and distributions.
Classical Probability
P(A) = favourable/total
Equally likely outcomes
Complement
P(A') = 1 β P(A)
P(A) + P(A') = 1 always
Addition Rule
P(AβͺB) = P(A)+P(B)βP(Aβ©B)
Subtract intersection to avoid double-counting
Independent Events
P(Aβ©B) = P(A) Γ P(B)
Also: P(A|B) = P(A)
Conditional Probability
P(A|B) = P(Aβ©B)/P(B)
Probability of A given B occurred
Multiplication Rule
P(Aβ©B) = P(A)Β·P(B|A)
Also = P(B)Β·P(A|B)
Total Probability
P(A) = Ξ£ P(Bα΅’)P(A|Bα΅’)
Bα΅’ exhaustive & exclusive
Bayes' Theorem
P(Bβ|A) = P(Bβ)P(A|Bβ)/P(A)
P(A) from total probability
Binomial P(X=r)
βΏCα΅£ pΚ³ qβΏβ»Κ³
q = 1βp; n trials; p = success prob
Binomial Mean/Var
ΞΌ = np; ΟΒ² = npq
Ο (SD) = β(npq)
Worked Examples
5 problems β classical probability to Bayes' to binomial.
EasyTwo dice rolled β find P(sum = 7)βΎ
Two fair dice are rolled. Find the probability that the sum is 7.
1
Total outcomes = 6 Γ 6 = 36
2
Favourable: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) = 6 outcomes
β P(sum = 7) = 1/6
EasyFind P(AβͺB) given P(A)=0.4, P(B)=0.3, P(Aβ©B)=0.1βΎ
Find P(A βͺ B) if P(A) = 0.4, P(B) = 0.3, P(A β© B) = 0.1.
1
P(AβͺB) = P(A) + P(B) β P(Aβ©B) = 0.4 + 0.3 β 0.1 = 0.6
β P(A βͺ B) = 0.6
MediumBag has 3 red, 4 blue balls. Two drawn without replacement. P(both red)?βΎ
A bag has 3 red and 4 blue balls. Two balls are drawn without replacement. Find P(both red).
2
P(2nd red | 1st red) = 2/6 = 1/3
3
P(both red) = (3/7) Γ (1/3) = 3/21 = 1/7
β P(both red) = 1/7
EAPCET LevelBayes' Theorem: factory output from two machinesβΎ
Machine A makes 60% of output with 2% defects; Machine B makes 40% with 5% defects. A defective item is found β what's the probability it came from Machine A?
2
P(defect|A) = 0.02, P(defect|B) = 0.05
3
P(defect) = 0.6Γ0.02 + 0.4Γ0.05 = 0.012 + 0.020 = 0.032
4
P(A|defect) = 0.6Γ0.02/0.032 = 0.012/0.032 = 3/8 = 0.375
β P(Machine A | defective) = 3/8 = 37.5%
Trap QuestionA fair coin tossed 5 times β P(at least one head)?βΎ
Find P(at least one head) when a fair coin is tossed 5 times. β οΈ Students compute directly instead of using complement.
1
Direct approach: P(exactly 1H) + P(2H) + ... + P(5H) = 5 separate calculations. Very slow.
2
Complement approach: P(at least 1H) = 1 β P(no heads) = 1 β P(all tails)
3
P(all tails) = (1/2)β΅ = 1/32
4
P(at least 1 head) = 1 β 1/32 = 31/32
β P(at least one head) = 31/32 (use complement: 1 β P(none))
Mistake DNA
4 probability errors from EAPCET distractor analysis.
β
P(AβͺB) = P(A) + P(B) Without Subtracting P(Aβ©B)
The addition rule requires subtracting the intersection to avoid double-counting overlapping outcomes.
β Wrong
P(AβͺB) = 0.4 + 0.3 = 0.7 β
(if P(Aβ©B) = 0.1,
this overcounts)
β Correct
P(AβͺB) = P(A)+P(B)βP(Aβ©B)
= 0.4+0.3β0.1 = 0.6 β
Subtract intersection
Only if A and B are mutually exclusive (P(Aβ©B)=0) does P(AβͺB) = P(A)+P(B). Always check mutual exclusivity.
π
Confusing Independent Events with Mutually Exclusive Events
Mutually exclusive: P(Aβ©B) = 0 (can't both happen). Independent: P(Aβ©B) = P(A)ΓP(B) (occurrence of one doesn't affect the other).
β Wrong
'A and B are mutually
exclusive β they are
independent' β
(nearly the opposite!)
β Correct
Mutually exclusive: Aβ©B = β
β
Independent: P(Aβ©B)=P(A)P(B) β
If P(A),P(B)>0: these
conditions can't both hold
If A and B are mutually exclusive with P(A)>0 and P(B)>0, then P(Aβ©B)=0 β P(A)P(B) β they are NOT independent.
π²
Not Using the Complement Method for 'At Least One'
'At least one' problems solved directly require many cases. Using complement (1 β P(none)) is always faster.
β Wrong
P(at least 1 six in 3 rolls):
P(1)+P(2)+P(3 sixes)
= 3 separate calculations β
β Correct
1 β P(no sixes)
= 1 β (5/6)Β³
= 1 β 125/216 = 91/216 β
One step, no cases
Whenever a question has 'at least one', 'at least once', or 'more than zero': use complement P = 1 β P(none/zero).
π
Binomial Distribution: Wrong q Value
q must be 1 β p. If p = 0.3, then q = 0.7. Students sometimes use q = p or forget it entirely.
β Wrong
P(X=2) with p=0.3, n=5:
β΅Cβ (0.3)Β² (0.3)Β³ β
(used p instead of q)
β Correct
q = 1 β 0.3 = 0.7 β
P(X=2) = β΅Cβ(0.3)Β²(0.7)Β³
= 10Γ0.09Γ0.343
= 0.3087 β
In binomial P(X=r) = βΏCα΅£ pΚ³ qβΏβ»Κ³: p = probability of success, q = 1βp = probability of failure. The exponents sum to n: r + (nβr) = n. β
Chapter Intelligence
Probability builds on P&C and leads to Statistics β master the fundamentals first.
EAPCET Weightage (2019β2024)
Conditional probability~7 Total probability theorem~3 High-Yield PYQ Patterns
P(AβͺB) using addition ruleConditional P(A|B) calculationBayes' theorem 2-machine problemBinomial mean and varianceP(at least one) using complementIndependent event identificationRandom variable expectation E(X)
Exam Strategy
- 'At least one' β always use complement: P = 1 β P(none). This is faster for every such problem.
- Bayes' theorem: set up a table with prior probabilities P(Bα΅’) and likelihoods P(A|Bα΅’). Compute the joint P(Bα΅’ β© A) = P(Bα΅’)ΓP(A|Bα΅’), then divide by their total.
- Binomial questions: identify n (trials), p (success probability per trial), and required r (number of successes). Apply P(X=r) = βΏCα΅£ pΚ³ qβΏβ»Κ³ directly.
- Independent vs mutually exclusive: if A and B are non-empty and mutually exclusive, they cannot be independent (knowing one gives info about the other).
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