| --- |
| source: ncert |
| topic: probability |
| class_level: class_12 |
| chapter: ch13 |
| difficulty: advanced |
| --- |
| |
| Class 12 | Ch13: Probability |
| CONDITIONAL PROBABILITY: |
| P(A|B)=P(A∩B)/P(B) where P(B)≠0. |
| P(A∩B)=P(A|B)*P(B)=P(B|A)*P(A). |
| MULTIPLICATION THEOREM: |
| P(A∩B)=P(A)*P(B|A). P(A∩B∩C)=P(A)*P(B|A)*P(C|AB). |
| INDEPENDENT EVENTS: |
| A and B independent: P(A∩B)=P(A)*P(B). P(A|B)=P(A). |
| A,B,C independent: P(A∩B)=P(A)P(B), P(B∩C)=P(B)P(C), P(A∩C)=P(A)P(C), AND P(A∩B∩C)=P(A)P(B)P(C). (All four conditions needed.) |
| TOTAL PROBABILITY THEOREM: |
| If B1,B2,...,Bn are mutually exclusive and exhaustive events: |
| P(A)=P(A|B1)*P(B1)+P(A|B2)*P(B2)+...+P(A|Bn)*P(Bn). |
| BAYES' THEOREM: |
| P(Bi|A)=P(A|Bi)*P(Bi) / [P(A|B1)*P(B1)+...+P(A|Bn)*P(Bn)]. |
| Prior probability: P(Bi). Posterior probability: P(Bi|A). |
| RANDOM VARIABLE AND PROBABILITY DISTRIBUTION: |
| Random variable X: function from sample space to real numbers. |
| Mean (Expected value): E(X)=mu=sum(xi*P(xi)). |
| Variance: Var(X)=E(X^2)-[E(X)]^2=sum(xi^2*P(xi))-mu^2. |
| SD=sqrt(Var(X)). |
| BERNOULLI TRIALS AND BINOMIAL DISTRIBUTION: |
| n independent trials, each with probability p of success. |
| P(X=r)=nCr*p^r*(1-p)^(n-r)=nCr*p^r*q^(n-r) where q=1-p. |
| Mean=np. Variance=npq. SD=sqrt(npq). |
| SOLVED EXAMPLES: |
| Example 1: Bag A: 3R,5B. Bag B: 4R,6B. One bag chosen randomly, red ball drawn. Find P(bag A). |
| P(A)=P(B)=1/2. P(R|A)=3/8. P(R|B)=4/10=2/5. |
| P(R)=P(R|A)P(A)+P(R|B)P(B)=(3/8)(1/2)+(2/5)(1/2)=3/16+1/5=31/80. |
| P(A|R)=P(R|A)P(A)/P(R)=(3/16)/(31/80)=(3/16)*(80/31)=15/31. |
| Example 2: Binomial X~B(10,0.4). Find P(X=3). |
| P(X=3)=10C3*(0.4)^3*(0.6)^7=120*0.064*0.0279936≈0.2150. |
| Mean=10*0.4=4. Variance=10*0.4*0.6=2.4. |
| COMMON MISTAKES: |
| Bayes theorem: denominator is TOTAL probability P(A) using all branches. |
| Binomial: trials must be INDEPENDENT with CONSTANT probability p. |
| E(X^2) ≠ [E(X)]^2. Variance=E(X^2)-[E(X)]^2. |
