## Modern Algebra Chapter 2 Probability

**Chapter 2 Probability Introduction**

**Probability:-**

- In our daily life, we have to face some events, the results of which are not certain, or it is impossible to guess about the certain results of that events.
- The circumstances that may evolve before the event or during the event or in the future, can not be forecasted accurately or certain results of the events can not be assumed correctly.
- It can be forecasted the results of the events only by assumption with the help of some logical arguments, that may or may not be wrong.
- Such as observing the clouds in the sky, we can assume that rain may take place or there is a probability of rain.
- Here, our assumption may be correct or incorrect, i.e., it may rain or may not rain.
- Let us through an unbiased dice.
- Then as a result of this event, we can say that any one of the numbers 1, 2, 3, 4, 5, and 6 may come.
- If we assume that 4 will come, then 4 may come or may not come.
- So, we have to say that there is a possibility or probability of coming 4 as a result.
- Similarly, the event of throwing an unbiased coin upwards is included in the probability theory.
- Here, either head or tail may occur as a result.
- At present, probability theory is widely used in the business world, statistics, preparation of budgets, determination of principles in government and non-government organizations, and also to forecast the demands of goods.
- Before the discussion of probability theory, we shall have some pre-knowledges about some concepts of it.
- In the following, these conceptions have been discussed.

**Read and Learn More WBBSE Solutions For Class 9 Maths**

## Chapter 2 Probability Random Experiment

**Random Experiment:-**

- The experiments or observations in which there is a clear conception about which events may occur or what results may come, but for a certain experiment of observation, it is impossible to say about the certain results, are known as random experiments or observations.
- Probability theory is always dealt with random experiments or observations.
- Again, if an unbiased dice is thrown randomly in the ground, we know that the result is either 1 or 2 or 3 or 4 or 5, or 6.
- But it is impossible to say about the certainty of the result.
- Similarly, in a football match for a team, the result is any one of the three, win, loss, or draw.
- So it is also an example of a random experiment.

**Events:**

- Any result related to a random experiment is called an event.
- For example, the throwing of an unbiased coin is either head or tail.
- Here, the occurrence of a head or tail in the coin throwing is an event of a random experiment.
- Similarly, each of the results 1, 2, 3, 4, 5, and 6, when a dice is rolled, is also an event.

## Chapter 2 Probability Mutually Exclusive Events

**Probability Mutually Exclusive Events:-**

- If two or more than two events are related in such a way that they can not occur together, then the events are called mutually exclusive events.
- If two events A and B are mutually exclusive, then \(A \cap B=\phi \text { or } P(A \cap B)=0, \text { where } \phi\) denotes the impossible event.
- Clearly, simple events are always mutually exclusive events.
- The odd and even results of throwing a die are mutually exclusive events.
- But in the same experiment, even results and the multiple of three results are not mutually exclusive events, because if the result be 6, then it is simultaneously even and a multiple of 3.

## Chapter 2 Probability Impossible Event And Certain (Or Sure) Event

**Impossible Event And Certain (Or Sure) Event:-**

- In any random experiment, we can think of or observe such an event that can never occur, this event is called an impossible event.
- Impossible events are generally denoted by P (Φ) = 0, i.e., the probability of an impossible event is zero.
- For example, let there are 4 red and 3 white balls in a box and 1 ball is withdrawn randomly.
- The ball will be green is an impossible event, since it can never occur.
- Similarly, to occur 7 in a dice throwing is also an impossible event.
- Again, in any random experiment, if we think or observe that an event must always occur in every case, then the event is called a certain or sure event.
- For example, in the random experiment of throwing a coin, the occurrence of a head or tail is a sure event.
- Sure events are usually denoted by S and the probability of a sure event is 1, i.e., P (S) = 1.

## Chapter 2 Probability Complementary Event

**Complementary Event:-**

- In any random experiment, the negative result of a certain event is called the complementary event.
- For example, the complementary event of the event head is not to occur head, i.e., the occurrence of a tail.
- In the rolling of a dice, the complementary event of the event occurrence of even is not to occur even result, i.e., the occurrence of odd results.
- The complement of an event A is usually denoted by \(\mathrm{A}^c \text { or } \mathrm{A}^{\prime} \text { or } \overline{\mathrm{A}} \text {. }\)

## Chapter 2 Probability Exhaustive Events

**Exhaustive Events:-**

- If two or more two events related to a random experiment be such that at least one of them must occur in the result of the experiment, then these events are known as exhaustive.
**For example,**in the random experiment of throwing an unbiased coin, if A and B are two events (where A occurrence of the head and B is = the occurrence of the tail), then the events A and B are said to be exhaustive since either A or B must occur in the result.

## Chapter 2 Probability Sample Space Or Event Space

**Sample Space Or Event Space:-**

- If E be a random experiment, then the simple events related to E are called simple point or event point.
- The set of all these event points which are possible in the experiment is called the sample space or event space of experiment E.
- Sample space is generally denoted by S.
- Thus, the universal set of all the events in an experiment is called the sample space or event space.

## Chapter 2 Probability Classical Or Mathematical Definition Of Probability

**Classical Or Mathematical Definition Of Probability:-**

- Let in a random experiment E, there is n (S) number of equally probable or mutually symmetrical sample points in the sample space S of E.
- If amongst the points in (A) number of points is included in the event A, then the ratio \(\frac{m(A)}{n(S)}\) is called the probability of the event A and it is denoted by P (A).
- ∴ \(\mathrm{P}(\mathrm{A})=\frac{m(\mathrm{~A})}{n(\mathrm{~S})} \quad \text { or, } \mathrm{P}(\mathrm{A})=\frac{\text { Number of event points equally probable related to } \mathrm{A} \text {. }}{\text { Number of total event points in the sample space. }}\)

## Chapter 2 Probability Notations

**Probability Notations:-**

- Let the sample space of the random experiment E be S and A and B are two events related to E.
- Then, P (A) denotes the probability of the event A.
- \(\left.\mathrm{P} \cdot\left(\mathrm{A}^c\right) \text { [or, } \mathrm{P}\left(\mathrm{A}^{\prime}\right) \text { or; } \mathrm{P}(\overline{\mathrm{A}})\right]\) denotes the probability of non-occurrence of event A.
- \(\mathrm{P}(\mathrm{A} \cup \mathrm{B})[\text { or, } \mathrm{P}(\mathrm{A}+\mathrm{B})]\) denotes the probability of occurrence of at least one of A and B, i.e., either A or B or both A and B occurs.
- \(P(A \cap B)[\text { or } P(A B)]\) denotes the probability of occurrence of both events A and B together.

## Chapter 2 Probability Some Theorems

**Theorem-1:** The probability of impossible events is zero, i.e., P (Φ) = 0.

**Theorem-2:** If A is an event, then \(\mathrm{P}\left(\mathrm{A}^c\right)=1-\mathrm{P}(\mathrm{A}), \text { where } \mathrm{A}^c\), where A is the complement of the event A.

**Theorem-3: **For an event A, 0 ≤ P (A) ≤ 1.

**Theorem-4: **

1. If A and B be two mutually exclusive events, then, \(P(A \cup B)=P(A)+P(B)\).

2. If A and B are not mutually exclusive events, then \(P(A \cup B)=P(A)+P(B)-P(A \cap B)\).

**Question 1. If E is the random experiment of throwing an unbiased coin and S is its sample space and if H and T denote the events of occurrence of head and tail respectively, then**

**1. If one coin is thrown, then S = {H, T}, here in sample space there are only two event points H and T.**

**2. If two coins are thrown simultaneously or one coin is thrown two times, then S = (HH, HT, TH, TT) and there are 2² = 4 event points in the sample space. S. **

**3. If three coins are thrown simultaneously or one coin is thrown three times, then find the sample space of E.**

**Solution: **

**Given**

**Three coins are thrown simultaneously or one coin is thrown three times.**

S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT). Here, there are 2³ = 8 sample points in the sample space S.

Let A denotes the event of the occurrence of the head at first and B denotes the event of the occurrence of two heads, then A= {HHH, HHT, HTH, HTT)

and B = {HHT, HTH, THH).

Clearly, \(A \cup B\) = {HHH, HHT, HTH, HTT, THH) and \(A \cap B\) = {HHT, HTH}

Again, if X denotes the event of non-occurrence of any head, then X occurrence of at least one head= complement of X = {TTT)}

= X_{c} = S – X

= {HHH, HTH, HHT, HTT, THH, THT, TTH}

If m coins are thrown simultaneously or one coin is thrown m-times, then there are 2m sample points in the sample space.

**Question 2. If E is a random experiment of rolling a dice and S be its sample space, then if one dice is rolled-**

**1. S = {1, 2, 3, 4, 5, 6), here the number of event points is 6.**

** 2. If two dice are rolled, find the sample space.**

**Solution:**

**Given**

**E is a random experiment of rolling a dice and S be its sample space.**

The required sample space = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4). (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)).

Here, in the sample space, there are 6² = 36 event points.

Similarly, if 3 dice are rolled, then there will be 6³ = 216 even points in the sample space.

If the number of dice is n, then the number of event points = 6^{n} in the sample space.

**Question 3. If E is a random experiment of counting the number of telephone calls in a telephone line after a regular period of time and if S is its sample space, then**

**Solution:**

**Given**

**E is a random experiment of counting the number of telephone calls in a telephone line after a regular period of time.**

**S is its sample space.**

S = {1, 2, 3, 4, 5, 6).

Here, there will have an infinite number of countable event points.

**Question 4. Let the heights of a group of students be more than 4 feet and less than 6 feet. ****Now, if E is the random experiment of measuring the heights of the students and if S is the sample space, then **

**Solution:**

**Given**

**The heights of a group of students be more than 4 feet and less than 6 feet.**

**E is the random experiment of measuring the heights of the students.**

**S is the sample space**

S = {x: 4 < x <6}

Here, the height of a certain student may be a real positive number between 4 feet and 6 feet.

Therefore, there will have infinitely many numbers of uncountable event points.

Clearly, in the given examples (1) and (2), there is a finite number of event points.

In example (3), there is an infinite number of countable event points and in example-(4), there is an infinite number of non-countable events.

Therefore, the sample spaces of examples (1), (2), and (3) are discrete, but in example-(4), it is continuous.