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Probability:

Probability is synonymous with possibility. It is a mathematical branch that deals with the occurrence of a random event. The value ranges from zero to one. Probability has been introduced in mathematics to predict the likelihood of events occurring. Probability is defined as the degree to which something is likely to occur. This is the fundamental probability theory, which is also used in the probability distribution, in which you will learn about the possible outcomes of a random experiment. To determine the likelihood of a single event occurring, we must first determine the total number of possible outcomes.

Probability Definition:

Probability is a measure of how likely an event is to occur. Many events are impossible to predict with absolute certainty. We can only predict the chance of an event occurring, i.e. how likely they are to occur, using it. Probability can range from 0 to 1, with 0 indicating an impossible event and 1 indicating a certain event. Probability for Class 10 is an important topic for students because it explains all of the fundamental concepts of this topic. The probability of all events in a sample space equals one.


For example, when we toss a coin, there are only two possible outcomes: Head OR Tail (H, T). When two coins are tossed, there are four possible outcomes: {(H, H), (H, T), (T, H), and (T, T)}.


Probability Formula:

According to the probability formula, the likelihood of an event occurring is equal to the ratio of the number of favorable outcomes to the total number of outcomes.

Probability of event to happen P(E) = Number of favorable outcomes/Total Number of outcomes


Probability Terms and Definition:


Some of the important probability terms are discussed here:


Types of Probability:

There are three major types of probabilities:

  • Theoretical Probability

  • Experimental Probability

  • Axiomatic Probability

Theoretical Probability:

It is based on the likelihood of something occurring. The reasoning behind probability is the foundation of theoretical probability. If a coin is tossed, the theoretical probability of getting a head is 12.


Experimental Probability:

It is based on the basis of observations of an experiment. The experimental probability can be calculated by dividing the total number of trials by the number of possible outcomes. For example, if a coin is tossed ten times and heads are recorded six times, the experimental probability of heads is 6/10, or 3/5.


Axiomatic Probability:

A set of rules or axioms that apply to all types is established in axiomatic probability. Kolmogorov established these axioms, which are known as Kolmogorov's three axioms. The axiomatic approach to probability quantifies the chances of events occurring or not occurring. This concept is covered in detail in the axiomatic probability lesson, which includes Kolmogorov's three rules (axioms) as well as various examples.

The likelihood of an event or outcome occurring based on the occurrence of a previous event or outcome is known as conditional probability.


Probability of an Event:

Assume that an event E can occur in r of n probable or possible equally likely ways. The event's likelihood of occurring or success is then expressed as;

P(E) = r/n

The likelihood that the event will not occur, also known as its failure, is expressed as follows:

(n-r)/n = 1-(r/n) P(E')

E' denotes that the event will not take place.

As a result, we can now say;

P(E) + P(E’) = 1

This means that the sum of all probabilities in any random test or experiment is 1.


What are Equally Likely Events?


When two events have the same theoretical probability of occurring, they are referred to as equally likely events. The outcomes of a sample space are said to be equally likely if they all have the same chance of occurring. For example, if you roll a die, the chance of getting 1 is 1/6. Similarly, the chance of getting all of the numbers from 2,3,4,5, and 6 at once is 1/6. As a result, the following are some examples of equally likely outcomes when rolling a die:

Getting 3 and 5 on a die throw

Obtaining an even and an odd number on a die

Getting a 1, 2, or 3 on a die roll

are equally likely events because their probabilities are equal.


Complementary Events:


The possibility that there will be only two outcomes, stating whether or not an event will occur. Examples of complementary events include a person coming or not coming to your house, getting a job or not getting a job, and so on. Essentially, the complement of an event occurring in the exact opposite probability that it will not occur. Here are some

more examples:


  • Today will either rain or not rain.

  • The student will either pass or fail the exam.

  • You either win or you don't.

Problems on Probability with solutions:


Example 1: A coin is thrown 3 times .what is the probability that at least one head is obtained? Sol: Sample space = [HHH, HHT, HTH, THH, TTH, THT, HTT, TTT] Total number of ways = 2 × 2 × 2 = 8. Fav. Cases = 7 P (A) = 7/8 OR P (of getting at least one head) = 1 – P (no head)⇒ 1 – (1/8) = 7/8


Example 2: Find the probability of getting a numbered card when a card is drawn from the pack of 52 cards. Sol: Total Cards = 52. Numbered Cards = (2, 3, 4, 5, 6, 7, 8, 9, 10) 9 from each suit 4 × 9 = 36 P (E) = 36/52 = 9/13


Example 3: There are 5 green and 7 red balls. Two balls are selected one by one without replacement. Find the probability that the first is green and the second is red. Sol: P (G) × P (R) = (5/12) x (7/11) = 35/132


Example 4: What is the probability of getting a sum of 7 when two dice are thrown? Sol: Probability math - Total number of ways = 6 × 6 = 36 ways. Favourable cases = (1, 6) (6, 1) (2, 5) (5, 2) (3, 4) (4, 3) --- 6 ways. P (A) = 6/36 = 1/6


Example 5: Three dice are thrown simultaneously. What is the probability of obtaining a total of 17 or 18?

Solution:

Three dice can be thrown in 6×6×6=216 ways.

A total of 17 can be obtained as (5,6,6),(6,5,6), (6,6,5).

A total of 18 can be obtained as (6,6,6).

Hence the required probability =4/216=1/54.


Example 6: In order to get at least once a head with probability ≥0.9, what is the number of times a coin needs to be tossed?

Solution:

Probability of getting at least one head in n tosses

=> 1 − (1/2)n ≥ 0.9

⇒(1/2)n ≤ 0.1

⇒ 2n ≥ 10

⇒ n ≥ 3

Hence the least value of n = 4.


Example 7: The three ships namely A, B, and C sail from India to Africa. If the ratio of the ships reaching safely is 2: 5, 3: 7 and 6: 11, then find the probability of all of them arriving safely.

Solution:

We have a ratio of the ships A, B and C for arriving safely are 2: 5, 3: 7 and 6: 11 respectively.

The probability of ship A for arriving safely = 2 / [2 + 5] = 2 / 7

Similarly, for B = 3 / [3 + 7] = 3 / 10 and for C = 6 / [6 + 11] = 6 / 17

∴Probability of all the ships for arriving safely = [2 / 7] × [3 / 10] × [6 / 17] = [18 / 595]


Example 8: If A and B are two events such that P (A) = 0.4 , P (A + B) = 0.7 and P (AB) = 0.2, then P (B) =

Solution:

Since we have P (A + B) = P (A) + P (B) − P (AB)

⇒ 0.7 = 0.4 + P (B) − 0.2

⇒ P (B) = 0.5.

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