Basic Theorems of Probability

You must have heard in news channel the reporter saying that there are some chances of rain tomorrow. Your friend, after an exam, says that he will score 90% in the exams Also, some news reports say one out of five children are suffering from malnutrition. What are all these? How can they predict and calculate all these? These are some of the examples of probability. What is probability? Let us get familiar with it.

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What is Probability?

The words like ‘certain’, ‘maybe’, ‘probably’, ‘never’ are related to the term probability. The literal meaning of probability is likely to happen. It is a measure for calculating the chances or the possibilities of the occurrence of a random event. In simple words, it calculates the chance of the favorable outcome amongst the entire possible outcome.

Mathematically, if you want to answer what is probability, it is defined as the ratio of the number of favorable events to the total number of possible outcomes of a random experiment. It is denoted by ‘p’. The probability of an event, say, E,

It is a number between 0 and 1. The number between 0 and 1 defines what is a probability. The number 0 shows the probability of an impossible event. In other words, the probability of not likely to happen or occur. The number 1 shows the probability of a sure event. Let us consider an example to understand what is the probability.

Before a cricket match starts, a coin is tossed to make a decision for the choice of batting first. Both the team captain chooses either head or tail. If the captain of team A selects head and of the team, B selects the tail. What is the probability that team A will win the toss?

Team A will win the toss if they get head in the toss. The chance of winning is the probability of occurrence of head = P(H) = ½. As the number of favorable cases = 1 (occurrence of a head) and the total number of outcome = 2 (head and tail). Now you are able to interpret what is probability in your terms.

Browse more Topics under Probability

• Introduction to Probability
• Probability of an Event
• Events and its Types
• Events and Its Algebra
• Independent Events
• Conditional Probability
• Multiplication Theorem on Probability
• Baye’s Theorem
• Random Variable and Its Probability Distribution
• Mean and Variance of Random Distribution
• Bernoulli Trials and Binomial Distribution

Terminologies to Understand What is Probability

Let us get familiar with some of the terms related to probability.

• Random Experiment: Those experiments, (conducted in identical situations), in which the outcomes are the same in each trail but its prediction is uncertain. The experiment in which the total numbers of outcomes are the same in all cases but the occurrence of the outcome is unpredictable is random.
• Outcome: The various possible results of a random experiment.
• Sample Space: The set of all possible outcomes is sample space. It is denoted by S.
• Sample Point: Each element of the sample space or each outcome of a random experiment.
• Event: A subset of a sample space is an event.
• Sure Event: The event which is certain to occur or the whole sample space.
• Impossible Event: An empty set is an impossible event.
• Complimentary Event: The non-happening of the event A is complimentary of the event A. It is denoted by A’. A’ = 1 – A. If A represents the success, then A’ represents the failure.

In a random experiment of tossing of a coin, the possible outcomes are Head and Tail. The sample space of this experiment is S = {H, T}. H represents Head and T represents Tail. An event E is getting a tail when we flip the coin.

The sure events in this experiment are getting the head or getting the tail of a coin. The impossible event is getting a number 6 when we toss a coin. The events of getting a head and a tail are complementary in this random experiment.

Basic Features of Probability

• The probability ranges from 0 to 1. 1: a certain result; 0: impossibility; and various in-between values measure the uncertainty.
• P[sum of all possible events]=1.
• P[sum of events]= Sum of probabilities of events.

Basic Theorems of Probability

There are some theorems associated with the probability. Let us study them in detail.

Theorem 1

The probability of the complementary event A’ of A is given by P(A’) = 1 – P(A).

Proof: The events A and A’ are mutually disjoint and together they form the whole sample space.

A ∪ A’ = S ⇒ P(A ∪ A’) = P(S) or, P(A) + P(A’) = P(S) = 1 ⇒ P(A’) = 1 − P(A).

Theorem 2

The probability of the impossible event is zero.

Proof: Let A be an impossible event and S be the sure event.  S = A’ and A = Φ.

P(Φ) = P(A) = P(S’) =  1 – P(S) = 1 − 1 = 0.

Theorem 3

If B subset A, then

1. P(A ∩ B’) = P(A) – P(B)
2. P(B) ≤ P(A).

Proof:

1. When B subset A, B and A ∩ B’ are mutually exclusive events. A = B ∪ (A ∩ B’). or, P(A) =  P[B ∪ (A ∩ B’)] = P(A) = P(B) + P(A ∩ B’) or, P(A ∩ B’) = P(A) – P(B).
2. P(A ∩ B’) ≥ 0 or, P(A) – P(B) ≥ 0 or, P(B) ≤ P(A).

Theorem 4

Let A be an event. Then 0 ≤ P(A) ≤ 1

Proof: Any event A which is a subset of S. A ⊂ S. P(A) ≤ P(S) or, P(A) ≤ 1. For any event A other than the impossible event, P(A) ≥ 0. Hence, 0 ≤ P(A) ≤ 1.

Theorem 5

If A and B are any two events and are not disjoint, then P(A ∪ B) = P(A) + P(B) – P(A ∩ B). A and B are the subsets of sample space S.

Proof: From the Venn diagram, we have

A ∪ B = A ∪ (A’ ∩ B). Here, A and A’ ∩ B are mutually disjoint.

 

Here, A’ ∩ B = B – (A ∩ B) and A ∩ B’ = A – (A ∩ B).

Therefore, P(A ∪ B) = P[A ∪ (A’ ∩ B)] = P(A) + P(A’ ∩ B) = P(A) + P(B) – P(A ∩ B).

This is the addition theorem of probability.

Theorem 6

If A and B are any two events such that P(A) ≠ 0 and P(B) ≠ 0. If A and B are independent events then P(A ∩ B) = P(A). P(B). This is the multiplication theorem of probability.

Solved Example for You

Question 1: A bag consists of 3 red balls, 5 blue balls, and 8 green balls. A ball is selected at random. Find the probability of

1. Getting a red ball.
2. Getting a green ball.
3. Not getting a blue ball.

Answer : Total number of the balls = 3 + 5 + 8 = 16.

1. Let R be the event of getting a red ball. The number of favorable outcome = 3. The required probability is P(R) = 3⁄16
2. Let G be the event of getting a green ball. The number of favorable outcome = 8. The required probability is P(G) = 8⁄16 = ½
3. Let B be the event of getting a blue ball. The number of favorable outcome = 5. The required probability of getting blue ball = 5⁄16. The probability of not getting a blue ball = 1 − P(B) = 1 – 5⁄16 = 11⁄16

Also, the event of not getting a blue ball is the same as getting a red or green ball. P(B’) = P(R) + P(G) = 3⁄16 + 8⁄16 = 11⁄16.

Question 2: Explain what is probability?

Answer: It refers to calculating the chance of a given event’s occurrence that is expressed as a number between 1 and 0. However, 5 can be considered to have equal odds of occurring or not occurring.

Question 3: What is the formula of probability?

Answer: It refers to the ratio of a number of favorable outcomes to the number of total possible outcomes. Moreover, it measures the possibility of an event in the following way: Suppose if P(A) > P(B) then event A is more likely to occur than event B. In the same way, if P(A) = P(B) the events A and B are likely to happen equally.

Question 4: What is the use of probability?

Answer: In mathematics, it is the likelihood that something will happen or not, such as drawing an ace from the deck of cards, picking a red candy from a bag of mixed color candies. Besides, in our daily life, it helps us to make decisions when you don’t know for sure what the outcome will be.

Question 5: State the two basic laws of probability?

Answer: The two basic law of probability is the law of multiplication and addition that we use for computing the probability of A and B, as well as the probability of A and B fro two given events A, B defined on the sample space.