Recalling Concepts of Probability

• We have come across many probability events by chance or in childhood.

• We find probability in many ways, like one of the methods is experimental.

• If we flip the coin over a number of times, we know that we are going to get head over half of the time.

• We always hear that new reporters are reporting about the chance of getting rain from the news.

• For example, he will say the chance of rain is 75% tomorrow due to a cyclone. It means out of all days, and it has rained 1 out of 75 of those days.

• Another way of finding probability is subjective, or we can say educational guess.

• Suppose someone asks you about who is going to win over the next cricket match(India and Pakistan). In that case, it is impossible to experiment because we don’t know whether the same two teams will play each other, the pitch and starts up should be the same or they will maintain the same field conditions.

• Since a lot of parameters are to be considered, it is impossible to say the probability by experiment.

• Someone who watches cricket or is familiar with the two-team player can easily guess there are chances of 75% to win only if the two team players under the same conditions India can win about 3 out of every four games. It can be told only by guess.

• The Educative guessers(subjective probability) may not be worth as much experimental probability. We will concern both in the concept of probability.

PROBABILITY DEFINITION IN MATH

• It is the measure of the event by chance or in childhood

• It can be predicted by the experimental or subjective gussets(theoretical ways).

• In math, the probability occurs between 0 to 1 in which 0 means no chance of probability; if it is 1 there may be certain events.

• In sample space, the probability of the events adds up to 1.

For example

• If we toss two coins, the expected outcome has three possibilities. Means head (HH), Tail head(T H), Tail tail(T T).

• If we toss one coin, there is only one possibility: Head (H) or Tail(T).

TYPES OF PROBABILITY

The concept of probability has four major types of probability. They are

1. Theoretical probability

2. Experimental probability

3. Axiomatic probability

BASIC CONCEPTS OF PROBABILITY

We will execute the experimental probability in dice, picking up the card and choosing hair color. Here are some of the terminology in the concept of probability.

Probability = Favorable outcomes during the trial/ total number of outcomes during the trail

                     = X/n

Where

x- a favorable outcome

n- number of outcomes

For example, if we take an example, if there is any possibility of rain tomorrow, the answer is yes or no.

TERMINOLOGY OF PROBABILITY THEORY

Here are some of the terminology that can help you understand the concept of probability better.

experiment – An experiment is conducted to know about the outcome during the trails

Sample space – All the possible outcomes we can expect in the trial.

For example

1. Tossing the coin twice, the sample space is (S) = (HH),(HT),(TT)

2. Rolling a dice, Sample space(S) = (1,2,3,4,5,6)

Sample point – Out of all outcomes expected for one results

For example, in the deck of cards

1. 4 spade at the same point

2. the king of heart ate the same point

Event – Single outcome of an experiment. For example – getting the tail while tossing a coin.

Outcome – possible outcome of a trial. For example- H is one of the possible outcomes of the tossing coin.

Complementary events – The not happening events. The complement of an event Q is the event, not Q(or Q’). For example, in the 52 deck cards, Q = draw as a Heart, then Q’ =don’t draw as a heart.

Impossible events – The event not taking place. For example, if we toss a coin, it is impossible to get head on simultaneously.

TRAIL

It indicated the experiment was conducted at random.

• Random experiment – Well-defined outcomes. For example, if we roll a dice, there might be the probability of getting any number between 1 to 6. But we don’t know which number is going to appear.

• Equally liked events – Two events are independent. For example, we have an equal chance of getting ahead or tail when we toss the coin.

• Exhaustive events – If the outcome is equal to the sample space

• Mutually experimental events – The event that can’t happen at the same time. For example, we can’t have a hot and cold climate simultaneously. We are either hot or cold.

BASIC PROBABILITY FORMULAS

The probability defines the happening events. In the concept of probability, the formula is given as below

Probability = Favorable outcomes during a trial/ total number of outcomes during a trial

                    = X/n or

P(X)            = number of favorable outcomes to X/ total number of possible outcomes.

                   = p(A)= [n(A)/n(S)]

Where

x- a favorable outcome

n- number of outcomes

n(S) = is the total number of events occurs during an trail

n(A) = is the favorable events

Probability formula with addition rule

Whenever the event occurs with the other two events

• P(S ∪ T) = P(S) + P(T) – P(S T)

Probability formula with the complementary rule

When one event is a complement to another event, this formula should have been used. If X is an event, then P(not an X)

• 1 -P(X) or P(X”) = 1 – P(X)

• P(X) + P(X”) = 1

Probability formula with the conditional rule

If the probability of occurrence of C is already known, then the D should be desired, Then P(D, given C) = P(C and D), P(C, given D). It can be vice versa in the case of event D.

P(D|C) = P(C D)/ P(C)

Probability formula with the multiplication rule

The two events intersect each other. Means event C and event D occur at the same time. The P ( C and D) = P(C).P(D)

P(C D) =P(C)>P(D|A)

Here are some of the examples for a better understanding of concepts of probability

EXAMPLE 1: Probability of getting less than 4 in the rolling a dice

Given sample space= (1,2,3,4,5,6)

Getting a number less than 4 is(1,2,3) then

n(S) = 6

n(A) = 3

By the probability formula

p(A)= [n(A)/n(S)]

p(A)= 3/6

p(A)= 1/2

PROBABILITY CONCEPT EXPLAINED

FINDING THE PROBABILITY EVENTS

In the experiment, the probability occurs between 1 and 0.

Event in probability 

If P(Z) represents the probability of the event Z, then

• P(Z) = 0 means there is no possibility that that event has happened.

• P(Z)= 1 than may be certain events

• 0≤ P(Z)≤ 1

COIN TOSS PROBABILITY

We can see how the concept of probability occurs in tossing the coin.

tossing a coin

Tossing a coin have two possible outcome

Sample space = Head(H) and tail(T)

P(H)= 1/2

P(T) = 1/2

DICE ROLL PROBABILITY

The concept of probability is also applied in the rolling of dice.

Rolling one dice

There may be six outcomes in the rolling of dice.

n(S) = {1,2,3,4,5,6}

P(even number) = the number of even is three{2,4,6}

so P(even) = 3/6 = 1/2

P( odd number) = the number of odd is three{1,3,5}

so P(odd) = 3/6 = 1/2

PROBABILITY OF DECK OF CARDS

The concept of probability is also applied in the deck of cards. There are 52 cards and divided into 4 groups. Each group has 13 cards.

The four groups are clubs, hearts, diamonds, spades. Where

• Spades and clubs are black

• heart and diamond are red

• There are a total of 26 black cards out of 52. P(black cards) = 26/52 = 1/2

• There are total 13 heart cards out of 52,P(Heart) = 13/52 = 1/4

• There are total 12 face cards out of 52,P(Heart) = 12/52 = 3/13

• The probability of drawing black card number 6 is P(6 black)= 2/52 = 1/26

• The probability of drawing a card number 6 is P(6)= 6/52= 3/26