Probability
Maths | Class 12 | Chapter 13 |
1. Where Does Probability Live Around You?
Probability is not just coins and dice. It runs real world.
Medical Test: If a disease is rare (1% have it) and test is 99% accurate, and you test positive, what is chance you actually have disease? Not 99%! Bayes theorem gives ~50%. Doctors use this daily.
Insurance: Insurance company calculates probability of accident, death, illness using past data to decide premium. Higher risk = higher premium.
Weather & Elections: 70% chance rain means out of 100 similar weather days, 70 had rain. Exit polls use probability to predict winner.
Quality Check & Games: Factory checks 5 items out of 100 to estimate defect probability. Games, lottery, cricket - all probability. Even your phone's spam filter uses Bayes.
So probability = measure of uncertainty, between 0 and 1. Closer to 1 = more likely.
Probability measures uncertainty 0 to 1. Real uses: medical test Bayes, insurance risk premium, weather 70% chance, elections, quality sampling, games, spam filter Bayes.
Probability 0 to 1 measures uncertainty. Uses: medical Bayes, insurance premium, weather forecast, elections, quality check, games, spam filter.
2. Conditional Probability - Given That Something Happened
Normal probability P(A) = chance of A. Conditional P(A|B) = chance of A given that B already happened. Sample space shrinks to B.
Formula: P(A|B) = P(A ∩ B) / P(B), provided P(B)≠0. Read as probability of A given B = probability of both A and B happening divided by probability of B.
Example: Bag has 3 red, 2 blue. Draw two without replacement. First draw red. What is P(second red | first red)? After first red, bag has 2 red,2 blue left. So P = 2/4 = 1/2. Using formula: P(both red)=3/5*2/4=6/20=3/10, P(first red)=3/5, so P(second|first)= (3/10)/(3/5)=1/2.
Properties: 0 ≤ P(A|B) ≤1, P(S|B)=1 where S sample space, P(A' |B)=1-P(A|B). If A and B mutually exclusive, P(A|B)=0.
Think like filter: given B happened, only look inside B.
P(A|B)=P(A∩B)/P(B), P(B)≠0. Sample space reduces to B. Example without replacement. Properties: 0-1 range, P(S|B)=1, P(A'|B)=1-P(A|B).
P(A|B)=P(A∩B)/P(B). Conditional shrinks sample to B. P(S|B)=1, P(A'|B)=1-P(A|B).
3. Multiplication Theorem and Independent Events
From conditional formula we get multiplication theorem: P(A ∩ B) = P(A) * P(B|A) = P(B) * P(A|B). This is chain rule. For 3 events: P(A∩B∩C)=P(A)P(B|A)P(C|A∩B).
Example: Bag 4 red 6 blue, two draws without replacement. P(both red)= P(first red) * P(second red|first red) = 4/10 * 3/9 =12/90=2/15.
Independent Events: Two events independent if occurrence of one does not affect other. Like tossing two different coins. Then P(B|A)=P(B) and P(A|B)=P(A). So formula becomes P(A∩B)=P(A)P(B).
Check independence: If P(A∩B)=P(A)P(B) then independent. Also P(A|B)=P(A), P(B|A)=P(B).
Example: Coin toss A=first toss head (1/2), B=second toss head (1/2). Both head =1/2*1/2=1/4 independent.
Difference: Mutually exclusive means cannot happen together P(A∩B)=0. Independent means can happen together but don't affect. Mutually exclusive with non-zero probabilities are always dependent.
P(A∩B)=P(A)P(B|A)=P(B)P(A|B). For 3 events chain. Independent if P(A∩B)=P(A)P(B), then P(B|A)=P(B). Mutually exclusive P(A∩B)=0 vs independent product.
P(A∩B)=P(A)P(B|A). Independent if P(A∩B)=P(A)P(B). Mutually exclusive P(A∩B)=0 is dependent if probabilities non-zero.
4. Total Probability and Bayes Theorem - The Reversal
Partition: Events E1,E2,...,En partition sample space if they are mutually exclusive and their union is whole sample space.
Theorem of Total Probability: If B can happen via different partitions Ei, then P(B)= Σ P(Ei) * P(B|Ei) = P(E1)P(B|E1)+...+P(En)P(B|En). You add contributions from all paths.
Example: Bag I 3 red 1 white, Bag II 2 red 3 white. Pick bag randomly then draw ball. P(white)= P(Bag I)*P(white|Bag I)+P(Bag II)*P(white|Bag II)=1/2*1/4+1/2*3/5=1/8+3/10=17/40.
Bayes Theorem (inverse probability): We know P(B|Ei) but want P(Ei|B) - cause given effect. P(Ei|B)= P(Ei)P(B|Ei) / Σ P(Ej)P(B|Ej) = P(Ei)P(B|Ei)/P(B).
Example above: Given white drawn, what is P(it came from Bag I)? P(Bag I|white)= P(Bag I)P(white|Bag I)/P(white)= (1/2*1/4)/(17/40)=5/17.
Real life: Disease test example, spam filter - given word appears, what is P(spam|word).
Total: P(B)=Σ P(Ei)P(B|Ei). Bayes: P(Ei|B)=P(Ei)P(B|Ei)/P(B)=P(Ei)P(B|Ei)/Σ P(Ej)P(B|Ej). Inverse: effect → cause.
Total P(B)=Σ P(Ei)P(B|Ei). Bayes P(Ei|B)=P(Ei)P(B|Ei)/P(B). Reverses conditional cause|effect.
5. Random Variable and Probability Distribution
Random Variable (X): Function that assigns number to each outcome. Example: Toss 2 coins, X = number of heads. Outcomes: TT→0, TH→1, HT→1, HH→2. So X can be 0,1,2.
Probability Distribution: List of values xi and probabilities pi = P(X=xi) with conditions: pi ≥0 and Σ pi =1.
Example for 2 coins: X=0 prob 1/4, X=1 prob 2/4=1/2, X=2 prob 1/4. Table: x 0 1 2, p 0.25 0.5 0.25, sum=1.
This table defines distribution. You can plot as bar chart. If sum not 1, not valid distribution.
Types: Discrete (finite values like dice) and continuous (not in Class 12 NCERT now, only discrete).
Random variable X assigns number to outcome. Distribution table xi, pi=P(X=xi), pi≥0, Σ pi=1. Eg 2 coins X=0,1,2 with probs 1/4,1/2,1/4.
X = number from outcome. Distribution xi, pi=P(X=xi), Σ pi=1, pi≥0.
6. Mean, Expectation and Variance
Mean / Expectation E(X): Average you expect if you repeat many times. E(X)= Σ xi * pi. Weighted average.
Example: 2 coins X=0,1,2 probs 1/4,1/2,1/4. E(X)=0*1/4+1*1/2+2*1/4=0+0.5+0.5=1. So expected heads in 2 tosses is 1.
E(X²)= Σ xi² * pi. Example same: 0²*1/4+1²*1/2+2²*1/4=0+0.5+1=1.5.
Variance Var(X)= E(X²) - [E(X)]². Measures spread. Standard deviation = √Var.
Example: Var =1.5 -1²=0.5.
Properties: E(aX+b)=aE(X)+b, Var(aX+b)=a²Var(X). If you add constant, variance unchanged.
E(X)=Σ xi pi, E(X²)=Σ xi² pi, Var=E(X²)-[E(X)]², SD=√Var. E(aX+b)=aE+b, Var(aX+b)=a²Var.
Mean E(X)=Σ xi pi. E(X²)=Σ xi² pi. Var=E(X²)-[E(X)]². SD=√Var.
7. Bernoulli Trials and Binomial Distribution
Bernoulli Trial: Trial with only two outcomes Success (p) and Failure (q=1-p). Like coin head/tail, pass/fail.
Bernoulli repeated n times independently → Binomial. X = number of successes in n trials.
Formula: P(X=r)= nCr * p^r * q^{n-r}, r=0,1,...,n. Where nCr = n!/(r!(n-r)!).
Why? Choose r positions for success out of n (nCr ways), each success prob p^r, each failure q^{n-r}.
Example: Fair coin 4 tosses, p=1/2, n=4, P(2 heads)=4C2*(1/2)²*(1/2)²=6*1/4*1/4=6/16=3/8=37.5%.
Mean and Variance for Binomial: Mean μ = np, Variance = npq, SD=√npq.
Example above n=4,p=0.5 mean=2, variance=1.
Conditions: Fixed n, independent trials, same p each trial, two outcomes only.
Bernoulli: two outcomes p, q=1-p. Binomial n independent trials X=successes. P(X=r)=nCr p^r q^{n-r}. Mean=np, Var=npq. Conditions: fixed n, independent, same p.
Bernoulli p,q=1-p. Binomial P(X=r)=nCr p^r q^{n-r}, Mean=np, Var=npq, conditions fixed n independent same p.
8. Complete Formula Summary - One Page Revision
Final revision:
Conditional: P(A|B)=P(A∩B)/P(B). Multiplication: P(A∩B)=P(A)P(B|A).
Independent: P(A∩B)=P(A)P(B), P(A|B)=P(A).
Total: P(B)=Σ P(Ei)P(B|Ei). Bayes: P(Ei|B)=P(Ei)P(B|Ei)/P(B).
Random Variable: xi, pi=P(X=xi), Σ pi=1.
Mean E(X)=Σ xi pi, Var=E(X²)-E(X)².
Binomial: P(X=r)=nCr p^r q^{n-r}, Mean=np, Var=npq.
P(A|B)=P(A∩B)/P(B). P(A∩B)=P(A)P(B|A). Independent P(A∩B)=P(A)P(B). Total P(B)=Σ P(Ei)P(B|Ei). Bayes P(Ei|B)=P(Ei)P(B|Ei)/P(B). Distribution Σ pi=1. E(X)=Σ xi pi, Var=E(X²)-E². Binomial nCr p^r q^{n-r}, μ=np, Var=npq.
All: Conditional P(A|B)=P(A∩B)/P(B), Multiplication P(A∩B)=P(A)P(B|A), Independent P(A∩B)=P(A)P(B), Total P(B)=Σ P(Ei)P(B|Ei), Bayes P(Ei|B)=P(Ei)P(B|Ei)/P(B), E(X)=Σ xi pi, Var=E(X²)-E², Binomial nCr p^r q^{n-r}, Mean np Var npq.
Conditional: P(A|B)=P(A∩B)/P(B). Multiplication P(A∩B)=P(A)P(B|A)=P(B)P(A|B). Independent P(A∩B)=P(A)P(B).
Total Probability: P(B)=Σ P(Ei)P(B|Ei). Bayes: P(Ei|B)=P(Ei)P(B|Ei)/Σ P(Ej)P(B|Ej).
Random Variable: Distribution xi, pi=P(X=xi), Σ pi=1, pi≥0. Mean E(X)=Σ xi pi, E(X²)=Σ xi² pi, Var=E(X²)-[E(X)]².
Bernoulli: p success, q=1-p failure. Binomial: n trials X successes P(X=r)=nCr p^r q^{n-r}, Mean=np, Var=npq, SD=√npq.
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