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Probability Formula is used to determine the chances of an event to occur. Probability simply means possibility.
- It is a branch of arithmetic that deals with the occurrence of a random event.
- The value of probability is expressed from 0 to one.
- Probability has been introduced in Maths to be expecting how probable events are to happen.
- The meaning of probability is the quantity to which something is probable to happen.
This is the simple possibility theory, which is likewise used withinside the possibility distribution, in which the opportunity of effects for a random experiment.
Read Also: Statistics
Key Terms: Probability, Probability Tree, Event, Coin, Sample Space, Trail, Experiment, Head, Tail
What is Probability?
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Probability means the measure of the likelihood that an event will occur. Many of the events cannot be predicted with absolute guarantee. With it we can only predict the probability of events, i.e. the probability of their occurrence.
The probability can range from 0 to 1, where o indicates the event is impossible and 1 indicates a specific event. The sum of the probabilities of all events in the sample space is 1.
Example of Probability
If we toss a coin and get one, there are only two possible outcomes (H, T).. But if we toss two coins in the air, there could be three possibilities of events to occur, such as both the coins show heads or both show tails or one shows heads and one tail, i.e. (H, H), (H,T), (T, T).
Probability of an Event
Assume an event E can occur in x ways out of n probable or possible equally likely ways. Then the probability of happening of that event is expressed as;
P(E) = \(\frac{x}{n}\)
The probability that the event will not occur or is failure is expressed as:
P(E’) = (n - r)/n = 1- (r / n)
E’ – represents that the event will not occur or ‘not E’ is called complement of the event E.
Therefore, now we can say;
P(E) + P(E’) = 1
Or,
P(E’) = 1 – P(E)
This means that the total of all the probabilities in any random trail is equal to 1.
- Events: When the events have the same probability of happening, they are called equally likely events.
- Complementary Events: The possibility of coming only two outcomes which states that an event will occur or not.
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Probability Definitions
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Some important definitions of Probability are:
| Term | Definition | Example |
|---|---|---|
| Sample Space | The set of all possible outcomes to occur in any trial. | Tossing a coin, Sample Space (S) = {H, T} |
| Sample Point | It is one of the possible results. | In a deck of cards: -4 of hearts is a sample point. -Queen of clubs is also a sample point. |
| Experiment or Trial | A series of trails where the outcomes are always uncertain. | Tossing of a coin, selecting a card from a deck of cards and throwing a dice. |
| Event | It’s a single outcome of an trail. | Getting a Head while tossing a coin is an event. |
| Outcome | Possible result of a trail/experiment. | T is a possible outcome when a coin is tossed, where T is Tail. |
| Complimentary Event | The non-happening events. The complement of an event A, not A (or A’). | Standard 52-card deck, R = Draw a heart, then R’ = don’t draw a heart. |
| Impossible Event | The event cannot happen. | In tossing a coin, it is impossible to get both Head and Tail at the same time. |
Probability Formulas
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The probability formula is used to find the chance of an event to occur. To recall, the chance of an event happening is known as probability.
- A probability is a chance of prediction.
- When we expect that, let’s say, x be the possibilities of going on an event then on the equal time (1-x) are the possibilities for “now no longer occurring” of an event.
- Similarly, if the probability of an event happening is “t” and an independent probability is “j”, then the probability of both the events occurring is “tj”.
- We can use this formula to find the chances of any event happening.
Formula to Calculate Probability
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The formula to calculate the probability of an event is:
P(K) = Number of Favorable Outcome/total number of Favorable outcomes
Or,
P(K) = n(K) / n(S)
Were,
- P(A) is the probability of an event ‘A’
- n(A) is the number of favorable outcomes (outcome of interest)
- n(S) is the total number of events in the sample space
| Probability Formula List | |
|---|---|
| Probability Range | 0 ≤ P(A) ≤ 1 |
| Rule of Addition | P(A∪ B) = P(A) + P(B) – P(A∩B) |
| Rule of Complementary Events | P(A’) + P(A) = 1 |
| Disjoint Events | P(A∩B) = 0 |
| Independent Events | P(A∩B) = P(A).P(B) |
| Conditional Probability | P(A | B) = P(A∩B) / P(B) |
| Bayes Formula | P(A | B) = P(B | A).P(A) / P(B) |
Things to Remember
- Probability is a phrase that is widely used in both arithmetic and statistics which helps to know the possibility an event is going to occur.
- Probability can be defined as the measure of the likelihood of an event to take place.
- Probability also offers an estimate of any event's uncertainty.
- The formula of Probability is P(K) = Number of Favorable Outcome/total number of Favorable outcomes
- The Probabilty value can be expressed from 0 to one.
Also Read:
| Chapter-Related Articles | ||
|---|---|---|
| Experimental Probability | Elementary Event | Sum of Probabilities |
| Geometric Probability | Sure Event | Probability Impossible Events |
Previous Year Questions
- A number xx is chosen at random from the set {1,2,3,4,.....,100}[JEE Main 2014]
- If now a ball is drawn at random from the bag, then the probability that this [JEE Main 2018]
- Then the expected value..[JEE Main 2020]
- If he decides to throw the die either till he gets a five or a six or to a maximum of three throws….[JEE Main 2019]
- If one of these students is selected at random, then the probability that the student: [JEE Main 2019]
- If two different numbers are taken from the set {0,1,2,3,……,10}then the probability that their sum as well as absolute difference are both multiple of 4, is :..[JEE Main 2017]
- Then the probability that one of the boxes contains excatly 3 balls, is...[JEE Main 2015]
- The probability that the card was drawn from Box I is :...[JEE Main 2020]
- Then , the events A and B are….[JEE Main 2014]
- The probability that one person speaks Hindi only and the other speaks both Hindi and English is….[KEAM]
Sample Questions
Ques: Which of the given experiments have equally likely outcomes? Explain:
1. A driver tries to start a car. The car starts or may not start.
2. A player strives to shoot a football. He misses or shoots the football.
3. A trial is made to get a solution: a true-false question. The solution may be right or wrong.
4. A child is born. Whether it is a girl or a boy. (2 marks)
Ans: 1. This statement does not have a likely outcome, as the car may begin to operate or it may not.
- This statement also does not have a likely outcome as the player may shoot or miss the ball.
- This statement has a likely outcome, as the solution has to be either right or wrong.
- This statement has a likely outcome because the newborn child has to be a girl or a boy.
Ques: Pull a random card from a pack of cards. What is the probability that the card pulled has a feminine face? (2 marks)
Ans: A standard deck of cards has 52 cards.
Total number of outcomes = 52
Number of favorable events = 4×1 = 4(considering queen only)
Therefore, probability(P) = Number of Favorable Outcomes ÷ Total Number of Outcomes
= 4/52
= 1/13
Ques: If P(E) = 0.25, What is the probability of ‘not E’? (2 marks)
Ans: We already know that
P(E) + P(not E) = 1
P(E) = 0.25
So, P(not E) = 1 - P(E)
P(not E) = 1 - 0.25
Hence, P(not E) = 0.75
Ques: If 10 defective balls are accidentally mixed with 144 good ones. It is not possible to just look at a ball and tell whether or not it is defective. One ball is drawn out at random from this set of balls. Determining the probability that the ball pulled out is a good one. (2 marks)
Ans: Number of balls = Number of defective balls + Number of good balls
Therefore, Total number of pens = 144 + 10 = 154 pens
P(E) = (Number of favorable results) / (Total number of results)
P(picking a good ball) = 144/154 = 72/77 = 0.935
Ques: Dangerous fires are very rare around 1% but the smoke is fairly common around 10% due to barbecues, and 90% of dangerous fires make a smoke. (2 marks)
Ans: We can calculate the probability of dangerous Fire when there is smoke Bayes theorem:
P(Fire|Smoke) = (P(Fire)P(Smoke Fire))/P(Smoke)
= 1/10
= 9%
Ques: A group of fifteen individuals sits around a round table. What are the chances of two people sitting together? (2 marks)
Ans: 15 people may be seated in 14 different ways. The number of different ways two individuals may sit together is 13! 2!
The likelihood of two specific people sitting together is 13!2! / 14! = 1/7.
The event's odds are 6: 1.
Ques: A coin is tossed three times. What is the likelihood of obtaining at least one head? (2 marks)
Ans: Sample space = [HHH, HHT, HTH, THH, TTH, THT, HTT, TTT, HTT, TTT]
The total number of ways is 2 2 2 = 8. P (of obtaining at least one head) = 1 – P (no head) 1 – (1/8) = 7/8 OR P (of getting at least one head) = 1 – P (no head) 1 – (1/8) = â…ž
Ques: Two dice are rolled together. What is the likelihood that the number rolled on one of the dice is a multiple of the number rolled on the other? (2 marks)
Ans: The total number of instances is 62, which equals 36.
Because a dice number should be a multiple of the other, the options are (1, 1) (2, 2) (3, 3) (———) (6, 6) —- 6 ways
(1, 2) (1, 4) (4, 1) (1, 3) (3, 1) (1, 5) (5, 1) (6, 1) (1, 6) —- 10 ways
(2), (4), (4), (4), (4), (4), (4), (4), (4), (4), (4), (4), (4), (4), (4), (4), (4), (4), (4),
Cases that are favorable are = 6 + 10 + 6 = 22. As a result, P (A) = 22/36 = 11/18.
Ques: One card is taken out from a well-shuffled deck of 52 cards. State the probability that the card will-
1. be a king.
2. not be a king. (3 marks)
Ans:
Well-shuffling of cards ensures fairly possible outcomes.
- Card drawn is a king
There are a total of four kings in a deck of cards.
Let A be the event ‘the card is a king’.
The number of outcomes favorable to A = n(A) = 4
The number of possible results = Total number of cards n(S) = 52
Hence, P(A) = n(A) / n(S) = 4/52 = 1/13
- Card drawn is not a king
let B be the event ‘card drawn is not s king’.
The number of outcomes favorable to the event B = n(b) = 52 – 4 = 48
Hence, P(B) = n(B) / n(S) = 48/52 = 12/13
Ques: Determine the likelihood of receiving HEAD at least once while tossing a coin twice. (3 marks)
Ans: In this case, sample space (S) = HH, HT, TH, TT.
H stands for Head, while T stands for Tail.
As a result, favorable events (E) = HH, HT, TH.
As a result, n (S) = 4 and n (E) = 3.
We obtain the following results when we plug these values into the probability formula:
P = 34 + 0.75 = 0.75
As a result, the chances of receiving at least one HEAD while flipping a coin twice are 0.75.
Ques: Determine the probability that a leap year will have 52 Sundays. (3 marks)
Ans: A leap year can contain 52 or 53 Sundays. A leap year has 366 days, of which 52 are full weeks and the remaining two are days. These two days can now be (Sat, Sun) (Sun, Mon) (Mon, Tue) (Tue, Wed) (Wed, Thur) (Thu, Fri) (Friday, Sat).
So there are a total of seven scenarios, two of which are favorable (Sat, Sun) and two of which are unfavorable (Sun, Mon). So, P = 2 / 7 (53 Sundays)
P(52 Sundays) + P(53 Sundays) Equals 1 now.
As a result, P (52 Sundays) = 1 – P(53 Sundays) = 1 – (2/7) = (5/7)
Ques: Calculate the likelihood of receiving an odd number if a die is rolled. (3 marks)
Ans: Sample space (S) = {1, 2, 3, 4, 5, 6}
6 = n(S)
Let "E" represent the occurrence of receiving an odd number, E = 1, 3, 5
3 = n(E)
As a result, the probability of receiving an odd number is:
P(E) = (Number of positive outcomes)/ (Total number of outcomes)
n(E)/n n(E)/n n(E)/n n (S)
3/6 = 1/2
Ques: What are the most popular formulae for probability sums? (4 marks)
Ans: Understanding the Probability sums is easier with set theory. The following are some of the formulae that are typically used in these sums.
P(A') = 1 + P (A)
n(E)/n P(A)= n(E)/n (S)
P(A.A') = 0
P(A.B) + P(A'.B') = 1.
P(A'B) = P(B) - P(A'B) (A.B)
P(A.B’)= P(A)-P(A.B)
P(A+B)= P(AB’) + P(A’B) +P(A.B)
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