Probability Meaning, Theorems and Examples

Probability indicates the chance of an event happening between 0 and 1. CBSE board exam may ask definition-based questions related to If the probability is 0, it means that the event will not happen since it has zero possibility. In case the probability is 1, it indicates that the event will happen for sure since it has a 100% probability of occurring. Between 0 and 1, the certainty increases as it approaches 1.

Probability of an event = $\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

• Number of favorable outcomes indicates the probability of getting the required outcome.
• Total number of possible outcomes mean the probability of any outcome that may happen.

Let us consider an example to calculate the probability. Suppose, there is a card deck which consists of 52 cards. What is the probability of drawing the Ace of Spades?

Total possible outcomes: There are 52 possible outcomes since there are 52 cards in total.

Favourable outcome: Only one ace of spades is in the card deck, which means that there is only one favourable outcome.

Therefore,

P(Ace of Spades) = 1/52 = 0.0192 or 1.92%

You will have to follow the steps given below to calculate the probability which is important for the JEE MAIN entrance exam and IIT JAM exam. Let us take a look at each one by one since:

• Understanding the Problem: Say, you want to calculate the probability of number 6 appearing on the die, or you want to calculate the probability of number 2 appearing simultaneously on two dice. You need to define the sample space according to that. For example, the sample space for one die will be [1,2,3,4,5,6] 
• Count Favourable Outcomes: Now, we need to count the number of favourable outcomes for the event. Say, you want to count the number of times 6 appears on rolling a die. We know that there is only 1 favourable outcome for this.
• Total Possible Outcomes: In a die, there are 6 faces. So when we roll a die, there are 6 possible outcomes. In simple words, when we roll a die, any number from 1,2,3,4,5 to 6 can come.
• Use formula: Now, we will calculate probability using the following formula, i.e. probability of an event = $\frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}$   

Let us take a look at the different types of probability:

1. Theoretical Probability: This type of probability is based on reasoning and mathematical theory. It assumes that each outcome is equally likely.

Probability of an event =  $\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

2. Experimental Probability: This type of probability is based on experiments or observations, which is calculated from actual data. The probability formula here is as given below

Probability of an event =  $\frac{\text{Number of favorable outcomes}}{\text{Total number of trials}}$

Let us consider an example of flipping a coin 100 times. We need to calculate the probability of getting 58 heads. P(Heads) = 58/100 = 0.58

3. Subjective Probability: This type of probability is based on experience, personal judgement or expert opinion. It is estimated instead of being calculated mathematically. This probability type is used in decision-making in uncertain situations.

4. Classical Probability: This is a subset of theoretical probability in which every outcome is equally likely. In most cases, it is used interchangeably with the theoretical probability.
5. Conditional Probability: This type of probability of an event considers that an event has already occurred. It is based on updating probability as new information is included.

$P\left(A\right|B)=\frac{P\left(A\text{ and }B\right)}{P\left(B\right)}$

6. Joint Probability: This is the probability of two or more events that occur simultaneously. P(A and B)=P(A)×P(B∣A) Marginal Probability: This is the probability of an event occurring regardless of any other event.

Let us take a look at the theorems used in probability:

1. Addition Theorem (Probability of A or B): For any two events, A and B, probability of either A or B occuring is P(A∪B) = P(A) + P(B)−P(A∩B) If A and B are mutually exclusive which means P(A∩B)=0, then P(A∪B) = P(A) + P(B) Let us take an example. Here, the probability of drawing a heart or king from deck:

P(Heart) = 13/52 P(King) = 4/52 P (Heart and King) = 1/52 ​P (Heart or King)

2. Multiplication Theorem (Probability of A and B): Given that there are two events A and B. The probability of the occurence of these two events will be: P(A∩B) = P(A) × P(B∣A) If A and B are independent i.e. P(B|A) = P(B), then: P(A∩B) = P(A) × P(B) Let us consider an example. The probability of rolling four on a die and flipping a heads on a coin (i.e. both events are independent). P(4)=1/6 P(Heads)=1/2​ P(4 and Heads)=1/6×1/2=1/12

3. Bayes’ Theorem: This theorem updates the probability of a hypothesis (A), which is based on new evidence (B).

P(B) is calculated using the Law of Total Probability. Say 1% of the population has been tested for a disease, and this test is 99% accurate. If a person is tested positive for the disease, what is the possibility that the person has that disease?

Question 1: Find the probability of getting a six on rolling dice once?

Solution:

Sample Space = {1, 2, 3, 4, 5, 6}

No of favourable events = 1

Total no. of outcomes = 6

Thus, P = ⅙ (Answer)

Question 2: Draw a random card from a deck of cards. What is the probability that the card drawn is an ace?

Solution:

A standard deck of cards has 52 cards in total.

Therefore, the total no. of outcomes = 52

No. of favourable outcomes = 4 (one for each suit- 4 suits namely Spade, Club, Hearts Diamonds)

Probability = No. of Favourable Outcomes/Total No. of Outcomes = 4/52= 2/26= 1/13 (Answer)

Question 3: What is the probability of getting two consecutive heads if we flip a regular coin twice?

Solution:

Sample Space: (H, H), (H, T),(T, T)

No. of favourable events: 1

Total outcomes: 3

Required probability: ⅓ (Answer)

Question 4:Take the mean and mode of the following data: 2, 3, 5, 6, 10, 6, 12, 6, 3, 4. 

Solution: 

Complete number: 10 

Sum of all numbers: 2+3+5+6+10+6+6+12+3+7=60 

Mean = (sum of all numbers)/(Total number of items) 

Mean = 60/10 = 6 

Again, Number 6 appears 3 times, so Mode = 6