This GSEB Class 12 Commerce Statistics Notes Part 2 Chapter 1 Probability Posting covers all the important topics and concepts as mentioned in the chapter.
Probability Class 12 GSEB Notes
Random Experiment and Sample Space:
Chance:
An unknown element upon which the occurrence or non-occurrence of an event depends is called chance.
Random Event:
The event which depends on chance is called a random event. For example, head is obtained on tossing a balanced coin, a card drawn from a pack of 52 cards is king of spade.
Random Experiment:
The experiment which can be independently repeated under identical conditions and all its possible outcomes are known but it cannot be predicted with certainty which of the outcome will appear is called a random experiment.
Characteristics of Random Experiment:
- It can be repeated under identical conditions.
- Its all possible outcomes are known.
- It cannot be predicted which of the outcome will appear at the end of experiment.
- It results into a certain outcome.
Sample Space:
The set of all possible outcomes of a random experiment is called a sample space of that random experiment. It is denoted by ‘U’ or ‘S’.
- Finite Sample Space: A sample space which has countable or finite number of elements is called a finite sample space. “For example, sample space obtained by throwing a balanced die.
- Infinite Sample Space: A sample space which does not have countable elements, i. e., if sample space is not finite then that sample space is called an infinite sample space. For example, the sample space of an experiment of drawing a card from the pack of 52 cards until the ace of heart is not obtained.
Sample Points:
The elements of a sample space are called sample points.
Event:
Any subset of the sample space of a random experiment is called an event. It is denoted by letters A, B, C,…
1. Impossible Event:
- A special subset 0 or { } of the sample space U of any random experiment is called an impossible event.
- For example, an event of getting both H and T in a single toss of an unbiased coin is an impossible event.
2. Certain Event:
- A special subset U of the sample space U of any random experiment is called a certain event.
- For example, an event of getting H or T in a single toss of an unbiased coin is a certain event.
3. Intersection of two events A and B:
If A and B be any two events of the finite sample space U, then the event that ‘event A and event B both occur simultaneously’ is called the intersection of events A and B. It is denoted by the symbol A ∩ B.
Thus, A ∩ B = {x; x ∈ A and x ∈ B}
4. Union of two events A and B:
If A and B be any two events of the finite sample space U, then the event that ‘either event A or B or both occur together is called the union of two events A and B. It is denoted by the symbol A ∪ B. Thus, A ∪ B = {x; x ∈ A or x ∈ B or x ∈ A ∩ B}
5. Complementary Event:
If A be an event of the finite sample space U, then, the event that A does not occur is defined as the set of those elements (or outcomes) of sample space U, which are not in A is called the complementary event of A. It is denoted by the symbol A’, AÌ… or Ac.
Thus, A’ = {x; x ∉ A, x ∉ U}
6. Mutually Exclusive Events: If A and B be any two events of a finite sample space; U, then the event that ‘events A and B; cannot occur together, i.e., if A ∩ B = Φ, the events A and B are said to be mutually exclusive events.
7. Difference Events:
If A and B be any: two events of the finite sample space U, then the set of all those elements of U, which belong to event A but do not belong to event B is called the difference event of A and B. It is denoted by the symbol A – B or A ∩ B’. Similarly, the set of all those elements of U which belong to event B but do not belong to event A is called the difference event of B and A. It is denoted by the symbol B – A or B ∩ A’.
Thus, A – B = {x; x ∈ A and x ∉ B}
B – A = {x; x ∈ B and x ∉ A}
8. Exhaustive Events:
If U is a sample space and A and B are any two events and A ∪ B = U, then events A and B are said to be exhaustive events.
9. Mutually Exclusive and Exhaustive Events:
If A and B be any two events of a finite sample space U such that A ∪ B = U and A ∩ B = Φ then A and B are said to be mutually exclusive and exhaustive events.
10. Elementary Events:
The events consisting of only a single element of a sample space U are called elementary events. The elementary events are mutually exclusive and exhaustive events.
11. Equi-probable Events:
If there is no apparent reason to believe that out of one or more events of a random experiment, any one event is more or less likely to occur them the other events, then the events are called equi-probable events.
12. Favourable Outcomes:
If some elementary outcomes out of all the elementary outcomes of a random experiment indicate the occurence of an event A, then these outcomes are said to be favourable to the occurence of the event A.
Mathematical or Classical definition of Probability:
Suppose U is a sample space of a random experiment and the total number of exhaustive, mutually exclusive and equiprobable elementary outcomes of it are n. If m (0 ≤ m ≤ n) outcomes out of n outcomes are favourable to the occurrence of an event A then, the probability of the event A is defined as \(\frac{m}{n}\). If we denote the probability of event A by the symbol P (A) then,
- P (A) = \(\frac{m}{n}\). Thus,
- P (A) = \(=\frac{\text { Number of favourable outcomes of event } \mathrm{A}}{\text { Total number of mutually exclusive, exhaustive and equiprobable outcomes of sample space }}\)
Assumptions of Mathematical or Classical definition of Probability:
- The number of outcomes of the sample space is finite.
- The number of all possible outcomes of the sample space is known.
- The outcomes of the sample space are equiprobable.
Limitations of Mathematical or Classical definition of Probability:
- When the number of outcomes of the sample space is infinite, it cannot be used to find the probability of an event.
- If the total number of outcomes of the sample space is unknown, the probability of an event cannot be obtained.
- If all possible outcomes of the sample space are not equiprobable, the probability of an event cannot be obtained.
Law of Addition of Probability:
A and B are any two events of a finite sample space U. The probability that at least one of the two events A and B will occur is called the law of addition of probability. This rule is written as under :
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
- If A ∩ B = Φ, then law of addition of probability is written as under:
P(A ∪ B) = P(A) + P(B) - If A and B are mutually exclusive and exhaustive events, then
P(A ∪ B) = P(A) + P(B) = 1. - The law of addition of probability for three events A, B and C of a sample space is written as under:
P(A ∪ B ∪ C) = P(A) + P(B) + P(C) – P(A ∩ B) – P(A ∩ C) – P(B ∩ C) + P(A ∩ B ∩ C) - If A, B, C are mutually exclusive events, then P (A ∪ B ∪ C) = P(A) + P(B) + P(C).
- If A, B, C are mutually exclusive and exhaustive events, then
P(A ∪ B ∪ C) = P(A) + P(B) + P(C) = 1.
Conditional Probability:
1. Conditional Events: A and B are any two events of a finite sample space U. Under the condition ‘event A has occurred’ if event B occurs, then that event B is called the conditional event. It is denoted by the symbol B | A. Similarly, under the condition ‘event B has occurred’ if event A will occur then that event A is called the conditional event. It is denoted by the symbol A|B.
2. Law of Conditional Probability: A and B are any two events of a finite sample space U and P (A) > 0. The rule to obtain the probability of event B | A, the probability of occurrence of event B given that event A has already occurred, is called the law of conditional probability. This rule is written as under:
P(B|A) = \(\frac{P(A \cap B)}{P(A)}\) P(A) > 0
Similarly, probability of conditional event A | B is obtained by following formula:
P(A|B) = \(\frac{P(A \cap B)}{P(B)}\), P (B) > 0
Independent Events:
A and B are any two events of a finite sample space U. The event B is called independent of the event A, if the probability of occurrence of event A does not affect the probability of occurrence of the event B. This means, if P(A|B) = P(A) and P(B|A) = P(B), then A and B are called mutually independent events.
If A and B are independent events, then
- A and B’ are independent events.
- A’ and B are independent events.
- A’ and B’ are also independent events.
Law of Multiplication of Probability:
A and B are any two events of a finite sample space U. The rule of obtaining the probability of the event AnB which corresponds to the occurrence of two events A and B simultaneously, is called the rule of multiplication of probability. This rule is written as under:
- P (A ∩ B) = P (A) . P (B | A)
OR
P (A ∩ B) = P (B) . P (A|B) - For two independent events law of multiplication of probability is written as under:
- P(A ∩ B) = P(A) P(B)
- P (A’ ∩ B’) = P (A’) . P (B’)
- P(A ∩ B’) = P(A) – P(B’)
- P(A’ ∩ B) = P (A’) . P (B)
Selection with Replacement and without Replacement:
- Selection with Replacement: In any trial if the selection of a unit from the population is done by replacing the unit selected in the previous trial back to the population then it is called the selection with replacement.
- Selection without Replacement: In any trial if the selection of a unit from the population is done by not replacing the unit selected in the previous trial back to the population, then it is called the selection without replacement.
Statistical definition of Probability:
If a random experiment is repeated n times under identical conditions and out of n such trials, m trials are favourable to the occurrence of some event A, then the relative frequency is called the estimate of the probability of occurrence of event A, which is denoted by P (A). If the value of n is taken larger and larger, i.e., as n tends to infinity, the limiting value of the ratio is taken as the probability of occurrence of the event A. Symbolically,
Limitations:
- The infinite value of n cannot be taken in practice.
- The exact value of probability cannot be known using this definition.
Important Formulae:
1. Permutation:
nPr = n (n – 1) (n – 2)… (n – r + 1)
For example, 8P3 = 8 (8 – 1) (8 – 2)
= 8 × 7 × 6 = 336
For example, 8P3 = \(\frac{8 !}{(8-3) !}=\frac{8 \times 7 \times 6 \times 5 !}{5 !}\) = 336
- nP0 = 1 For example, 6P0 = 1
- nP1 = n For example, 6P1 = 6
- nPn = nPn-1 = n ! For example, 6P6 = 6P5 = 6 !
- n! = n (n – 1) (n – 2) (n – 3)… × 3 × 2 × 1
2. Identical Permutations:
Total permutations of n item = \(\frac{n !}{p ! q ! r !}\) where p, q, r are identical items.
3. Combination:
nCr = \(\frac{n !}{r !(n-r) !}\)
For example, 10C3 = \(\frac{10 !}{3 ! 7 !}=\frac{10 \times 9 \times 8}{3 \times 2 \times 1}\) = 120
- nCn = nC0 = 1 For example, 10C10 = 10C0 = 1
- nC1Â = nCn-1 = n For example, 10C1 = 10C9 = 10
- nCr = nCn-r For example, 10C3 = 10C10-3 = 10C7
4. Probability:
Symbols:
- U = Sample space
- Φ = Impossible event
- U = Certain event
- A’ = Complementary event of event A
- A ∩ B = Intersection of events A and B
- A ∪ B = Union of events A and B
- A – B = Difference of events A and B = A ∩ B’
- B-A = Difference of events B and A = A’ ∩ B
- P (A) = Probability of event A
- n = Elementary outcomes of random experiment
- m = Favourable elementary outcomes for happening of any event
Number of favourable elementary outcomes
1. Probability = \(=\frac{\text { Number of favourable outcomes of elementary outcomes } \mathrm{A}}{\text { Total number of elementary outcomes }}\)
For example, for a random experiment, if out of total n elementary outcomes, m elementary outcomes are favourable to the occurrence of the event A, then
P (A) = \(\frac{m}{n}\) where, 0 ≤ m ≤ n and 0 ≤ P(A) ≤ 1.
2. P(U) = 1 where, U = certain event
P (0) = 0 where, Φ = impossible event
3. P(A’) = 1 – P(A),
i.e., P(A) + P(A’) = 1
4. If A ⊂ B, then P(A) ≤ P(B).
5. Law of Addition:
- If A and B are any two events, then P (A ∪ B) = P(A) + P(B) – P(A ∩ B).
- If A and B are mutually exclusive events, then P(A ∪ B) = P(A) + P(B).
- If A, B and C are any three events, then P(A ∪ B ∪ C) = P(A) + P(B) + P(C) – P(A ∩ B) – P(B ∩ C) – P(A ∩ C) + P(A ∩ B ∩ C).
- If A, B and C are mutually exclusive, then P(A ∪ B ∪ C) = P(A) + P(B) + P(C).
- If events A and B are exhaustive and mutually exclusive, then P(A ∪ B) = P(A) + P(B) = 1.
- If events A, B and C are exhaustive and mutually exclusive, then P(A ∪ B ∪ C) = P(A) + P(B) + P(C) = 1.
6. Law of Conditional Probability: If A and B are any two events, then
P(A|B) = \(\frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{\mathrm{P}(\mathrm{B})}\) and
P(B|A) =\(\frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{\mathrm{P}(\mathrm{A})}\) where, P(A) > 0; P(B) > 0
7. Law of Multiplication: If A and B are any two events, then
P(A ∩ B) = P(A|B) – P(B) or P(A ∩ B) = P(B|A) – P(A).
- If A and B are mutually independent events, then P(A ∩ B) = P(A) – P(B)
- P(A’ ∩ B’) = P(A’)-P(B’)
- P(A’ ∩ B) = P(A’) -P(B)
- P(A ∩ B’) = P(A) P(B’)
8. Some Results :
P(A’ ∪ B’) = P(A ∩ B)’ = 1 -P(A ∩ B)
P(A’ ∩ B’) = P(A ∪ B)’ = 1 -P(A ∪ B)
- P(A – B) = P(A ∩ B’) = P(A) – P(A ∩ B) P(B – A) = P(A’ ∩ B) = P(B) – P(A ∩ B)
- If A ⊂ B, then P(A) ≤ P(B) and P(B – A) = P(B) – P(A).
- A ∩ B ⊂ A
∴ P(A ∩ B) ≤ P(A) - A ∩ B ⊂ B
∴ P(A ∩ B) ≤ P(B) - If A ⊂ (A ∪ B) and B ⊂ (A ∪ B), then
P(A)<(A ∪ B) and P(B)^(A ∪ B) - A = (A ∩ B) ∪ (A ∩ B’);
B = (A ∩ B) ∪ (A’ ∩ B) - (A ∩ B) ∩ (A ∩ B’) = Φ and (A ∩ B) ∩ (A’ ∩ B) = Φ
P(A) = P(A ∩ B) + P(A ∩ B’) and P(B) = P(A ∩ B) + P(A’ ∩ B) - 0 ≤ P(A ∩ B) ≤ P(A) ≤ P(AuB) ≤ P(A) + P(B)
9. Statistical definition of Probability:
