EASY
12th CBSE
IMPORTANT
Earn 100

Given that E and F are events such that PE=0.6,PF=0.3 and PEF=0.2, find PEF and PFE.

Important Points to Remember in Chapter -1 - Probability from NCERT MATHEMATICS PART II Textbook for Class XII Solutions

1. Elementary event:

If a random experiment is performed, then each of its outcomes is known as an elementary event.

2. Sample space of an experiment:

The set of all possible outcomes of a random experiment is called the sample space associated with it and it is generally denoted by S

3. Simple event and Compound event:

(i) A subset of the sample space associated to a random experiment is said to define a simple event if it has only one outcome.

(ii) A subset of the sample space associated to a random experiment is said to define a compound event if it has more than one outcome.

4. Favourable elementary events:

Let S be the sample space associated with a random experiment and A be an event associated to the experiment. Then elementary events belonging to A are known as favourable elementary events to the event A.

5. Probability of an event:

If there are n elementary events associated with a random experiment and m of them are favourable to an event A, then the probability of happening or occurrence of A is denoted by P(A) and is defined as the ratio mn.

6. Relationship between the probability of an event and its complement:

The sum of probability of an event and its complement is 1. So, given an event A, we have P(A¯)+P(A)=1.

7. Odds ratio: 

Number outcomes in the favour of event A be m and number of outcomes against the event A be n, then

(i) The odds in favour of occurrence of the event A are defined by m:n i.e., PA:PA¯

(ii) The odds against the occurrence of A are defined by n:m i.e., PA¯:PA.

8. Types of events and the relationship between their probabilities:

(i) Mutually exclusive events:

(a) Two or more events associated to a random experiment are mutually exclusive if they have no common outcomes.

(b) If two events A and B are mutually exclusive, then PAB=0. Similarly, if A, B and C are mutually exclusive events, then PABC=0.

(c) All elementary events associated to a random experiment are mutually exclusive.

(ii) Exhaustive events:

(a) Two or more events associated to a random experiment are exhaustive if their union is the sample space. i.e. events A1,A2,...,An associated to a random experiment with sample space S are exhaustive if A1A2 ...An=S.

(b) All elementary events associated to a random experiment form a system of exhaustive events.

(c) For any event A associated to a random experiment, A and A form a pair of exhaustive and mutually exclusive events.

(iii) Independent events:

Two events A and B associated to a random experiment are independent if the probability of occurrence or non-occurrence of A is not affected by the occurrence or non-occurrence of B.

NOTE: If A and B are two mutually exclusive events associated to a random experiment, then the occurrence of any one of these two prevents the occurrence of the other i.e., If A occurs, then PB=0 and if B occurs, then PA=0. It follows from this that mutually exclusive events associated to a random experiment are not independent and vice-versa.

9. Theorems of probability:

(i) Addition theorem of probability for two events:

If A and B are two events associated with a random experiment, then PAB=PA+PB-PAB

(ii) Addition theorem of probability for two mutually exclusive events: 

If A and B are mutually exclusive events, then PAB=0 and hence, PAB=PA +PB

(iii) Addition theorem of probability for three events:

If A, B, C are three events associated with a random experiment, then PABC=PA+PB+PC-PAB-PBC-PAC+PABC

(iv) Probability of occurrence of event A but not B:

Let A and B be the two events associated to a random experiment. Then,

(a) P(A¯B)=PB-PAB

(b) P(AB¯)=PA-PAB

(v) Probability of occurrence of exactly one of the two event:

(AB¯)(A¯B) represents the occurrence of exactly one of two events A and B. Its probability is given by    

(a) PAB¯A¯B=PA+PB-2PAB

(b) PA+PB-2PAB=PAB-PAB.

10. If A, B, C are three events, then

(i) P(At least two of A, B, C occur) =PAB+PBC+PCA-2PABC

(ii) P(Exactly two of A, B, C occur) =P(AB)+PBC+PAC-3PABC

(iii) P (Exactly one of A, B, C occurs) =PA+PB+PC-2PAB-2PBC-2PAC+3PABC.

11. Conditional probability: 

Let A and B be two events associated with a random experiment. Then, the probability of occurrence of event A under the condition that B has already occurred and P(B)0, is called the conditional probability and it is denoted by PAB

12. Conditional probability of independent events:

If A and B are independent events associated with a random experiment, then PAB=PA.

13. Probability of intersection of two events:

If A and B are two events associated with a random experiment, then P(AB)=PAPBA, if PA0 or, PAB=PBPAB, if PB0

14. Probability of intersection of n events: If A1, A2, ..., An are n events associated with a random experiment, then

PA1A2A3 ....An=PA1PA2A1PA3A1A2……. PAnA1A2....An-1

15. Properties of independent events:

If A and B are independent events associated with a random experiment, then

(i) A¯ and B are independent events.

(ii) A and B¯ are independent events.

(iii) A¯ and B¯ are also independent events.

16. Total Probability Theorem: 

Let S be the sample space and let E1, E2, ..., En be n mutually exclusive and exhaustive events associated with a random experiment. If A is any event which occurs with E1 or E2 or ... or En, then

PA=PE1PAE1+PE2PAE2+ ... +  + PEnPAEn or PA=r=1nPErPAEr

17. Baye’s theorem: 

Let S be the sample space and let E1, E2..., En be n mutually exclusive and exhaustive events associated with a random experiment. If A is any event which occurs with E1 or E2 or ... or En, then

PEiA=PEiPAEii=1nPEiPAEi, i=1, 2, , n

18. Random variable:

Let S be the sample space associated with a given random experiment. Then, a real valued function X which assigns each event w  S to a unique real number X (w) is called a random variable.

19. Probability distribution of a random variable:

If a random variable X takes values x1, x2, ...., xn with respective probabilities p1, p2,.... , pn then

X : x1 x2 x3 .... xn
P(X) : p1 p2 p3 .... pn

 where pi0 and pi=1 is known as the probability distribution of X.

20. If X is a random variable with the probability distribution

X : x1 x2 .... xn
P(X) : p1 p2 .... pn

then,

(i) P(Xxi)=PX=x1+PX=x2+  + P(X=xi)=p1+p2 + ....+ pi

(ii) P(X<xi)=PX=x1+PX=x2+  + PX=xi-1 = p1+p2 + ....+ pi-1

(iii) PXxi=PX=xi+PX=xi+1+  + PX=xn = pi+pi+1 + .... + pn

(iv) PX>xi=PX=xi+1+PX=xi+2+  + PX=xn = pi+1+pi+2 + .... + pn

(v) PX>xi=1 PXxi

(vi) PX<xi=1-PXxi

(vii) P xiXxj=PX=xi+P(X=xi+1) + ... + PX=xj

(viii) P( xi<X<xj)=P(X=xi+1)+PX= xi+2 +  + PX=xj-1

21. Mean of a discrete random variable:

If X is a discrete random variable which assumes values x1, x2, x3, ...., xn with respective probabilities p1, p2, p3, ...., pn, then the mean X¯ of X is defined as X¯=p1x2+p2x2+p3x3+....+pnxn or, X¯=EX=i=1npi xi

22. Variance of a discrete random variable:

If X is a discrete random variable which assumes values x1, x2, x3, ...., xn with the respective probabilities p1, p2, p3, ...., pn, then variance of X is defined as Var X=p1x1-X¯2+p2 x2 X¯2+ . + pn xn  X¯2

 VarX=i=1np ixi-X¯2, where X-=i=1npixi is the mean of X.

Also, VarX=i=1npi xi2-i=1npixi2

23. Probability distribution function of a Binomial variate:

A random variable X which takes values 0,1,2,...,n, is said to follow binomial distribution if its probability distribution function is given byPX=r=Cr nprqn-r,r = 0,1,2,...,n, where p, q> 0 such that p + q=1.

24. Mean and Variance of a Binomial variate: 

The mean and variance of a binomial variate with parameters n and p are np and npq respectively.