Discrete Mathematics Probability - Discrete Mathematics

What is the concept of probability in Discrete Mathematics?

Finding a chance of occurrence of a particular event is known as probability. The study of random processes and outcomes is known as probability.

Probability is applied in different fields such as genetics, weather forecasting, stock markets, opinion polling etc.

What are the basic concepts of probability?

Two mathematicians Blaise Pascal and Pierre de Fermat, who were dealing with the problems related to chance has invented the the theory of probability in 17th century.

Random Experiment

An experiment is a random experiment, when the exact output cannot be predicted in advances and when all the possible outcomes are known. One of the examples of ransom experiment is tossing of a coin.

Sample Space

For an experiment, the set S of all possible outcomes in known as the sample space. The sample space for tossing of a coin is S={H,T}S={H,T}.

Event

The subset of the sample space is called as an event. Once the coin is tossed, getting head or tail is an event.

The chance of occurrence of a particular event is probability.

The probability varies from 0 to 1 as the chance of occurrence of the event varies from 0% and 100%.

What are the steps to find the probability?

The following are the steps to find the probability:

Step 1 – All the possible outcomes of the experiment are calculated.

Step 2 – The number of favourable outcomes of the experiment are calculated

Step 3 – The corresponding probability formula is applied.

Tossing a Coin

There are two possible outcomes for tossing a coin – Heads (H) or Tails (T). Hence total number of outcomes is 2.

The probability of getting a Head is 1/2 and the probability of getting a tail is on top is ½.

Throwing a Dice

There are six possible outcomes on the top, when a dice is thrown - 1,2,3,4,5,6.

The probability of any one of the numbers is 1/6

The probability of getting even numbers is 3/6 = 1/3

The probability of getting odd numbers is 3/6 = 1/3

Taking Cards From a Deck

From the pack of 52 cards, what is the probability of ace being drawn and what is the probability of diamond being drawn?

Total number of possible outcomes − 52

Outcomes of being an ace − 4

Probability of being an ace = 4/52 = 1/13

Probability of being a diamond = 13/52 = 1/4

What are the different axioms of Probability?

The axioms of probability are as follows:

  • The probability of an event always varies from 0 to 1. [0≤P(x)≤1]
  • The probability is 0 for impossible event and 1 for certain and possible event.
  • The events are said to be mutually exclusive or disjoint if the occurrence of one event does not influence the other.

If
are mutually exclusive/disjoint events, then

What are the different properties of Probability?

The following are the properties of probability:

  • •If there are two eventswhich are complementary, then the probability of the complementary event is −

  • For two non-disjoint events A and B, the probability of the union of two events −

P(A∪B)=P(A)+P(B)

•If an event A is a subset of another event B (i.e. A⊂B, then the probability of A is less than or equal to the probability of B. Hence, A⊂B implies P(A)≤p(B)

What is Conditional Probability?

Given that the event A has already occurred, the probability that the event B will occur is the conditional probability of the event B. This is denoted as P(B|A).

Mathematically − P(B|A)=P(A∩B)/P(A)

If event A and B are mutually exclusive, then the conditional probability of event B after the event A will be the probability of event B that is P(B).

Problem 1

For instance, 50% of the teenagers of a country own a cycle and 30% of all teenagers own a bike and cycle. Find the probability of the teenagers who own bike given that the teenagers own a cycle?

Solution

It is assumed that A is the event of teenagers owning only a cycle and B is the event of teenagers owning only a bike.

So, P(A)=50/100=0.5 and P(A∩B)=30/100=0.3 from the given problem.

P(B|A)=P(A∩B)/P(A)=0.3/0.5=0.6

Hence, the probability that a teenager owns bike given that the teenager owns a cycle is 60%.

Problem 2

Cricket is played by 50% of the class students, and 25% play both cricket and volleyball. Find the probability of students playing volleyball given that the student plays cricket.

Solution

It is assumed that A is the event of the students who play only cricket and B is the event of students who play only volleyball.

So, P(A)=50/100=0.5 and P(A∩B)=25/100=0.25 from the given problem.

Therefore the probability that the student plays volleyball given that the student plays cricket is 50%.

What is Bayes' Theorem?

Bayes’ Theorem – If A and B are two mutually exclusive events with P(A) being the probability of A and P(B) being the probability of B, then P(A|B) is the probability of A provided that B is true and P(B|A) is the probability of B, given that A is true. Hence the Bayes’ Theorem states that:

Application of Bayes' Theorem

•When the events of the sample space are mutually exclusive events.

•In situations where either for each of theAior P(Ai) and P(B|Ai) for each of the Ai is known.

Problem

For instance, three pen-stands are considered. In the first pen stand, there are 2 red pens and 3 blue pens. The second pen stand has 3 red pens and 2 blue pens. The third pen stand has 4 red pens and 1 blue pen. The probability of each pen stand getting selected is equal. If anyone pen is drawn, find the probability that it is a red pen?

Solution

LetAibe the event that ith pen-stand is selected.

Here, i = 1,2,3.

Since probability for choosing a pen-stand is equal, P(Ai)=1/3

Let B be the event that a red pen is drawn.

The probability that a red pen is chosen among the five pens of the first pen-stand,

The probability that a red pen is chosen among the five pens of the second pen-stand,

The probability that a red pen is chosen among the five pens of the third pen-stand,

According to Bayes' Theorem,

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