Probability is a branch of mathematics that deals with the happening of a random event. It is used in Maths to predict how likely events are to happen.
The probability of any event can only be between 0 and 1 and it can also be written in the form of a percentage.
The probability of event A is generally written as P(A).
Here P represents the possibility and A represents the event. It states how likely an event is about to happen. The probability of an event can exist only between 0 and 1 where 0 indicates that event is not going to happen i.e. Impossibility and 1 indicates that it is going to happen for sure i.e. Certainty
If we are not sure about the outcome of an event, we take help of the probabilities of certain outcomes—how likely they occur. For a proper understanding of probability we take an example as tossing a coin:
There will be two possible outcomes—heads or tails.
The probability of getting heads is half. You might already know that the probability is half/half or 50% as the event is an equally likely event and is complementary so the possibility of getting heads or tails is 50%.
Formula of Probability
Probability of an event, P(A) = Favorable outcomes / Total number of outcomes
Some Terms of Probability Theory
- Experiment: An operation or trial done to produce an outcome is called an experiment.
- Sample Space: An experiment together constitutes a sample space for all the possible outcomes. For example, the sample space of tossing a coin is head and tail.
- Favorable Outcome: An event that has produced the required result is called a favorable outcome. For example, If we roll two dice at the same time then the possible or favorable outcomes of getting the sum of numbers on the two dice as 4 are (1,3), (2,2), and (3,1).
- Trial: A trial means doing a random experiment.
- Random Experiment: A random experiment is an experiment that has a well-defined set of outcomes. For example, when we toss a coin, we would get ahead or tail but we are not sure about the outcome that which one will appear.
- Event: An event is the outcome of a random experiment.
- Equally Likely Events: Equally likely events are rare events that have the same chances or probability of occurring. Here The outcome of one event is independent of the other. For instance, when we toss a coin, there are equal chances of getting a head or a tail.
- Exhaustive Events: An exhaustive event is when the set of all outcomes of an experiment is equal to the sample space.
- Mutually Exclusive Events: Events that cannot happen simultaneously are called mutually exclusive events. For example, the climate can be either cold or hot. We cannot experience the same weather again and again.
- Complementary Events: The Possibility of only two outcomes which is an event will occur or not. Like a person will eat or not eat the food, buying a bike or not buying a bike, etc. are examples of complementary events.
Some Probability Formulas
Addition rule: Union of two events, say A and B, then
P(A or B) = P(A) + P(B) – P(A∩B)
P(A ∪ B) = P(A) + P(B) – P(A∩B)
Complementary rule: If there are two possible events of an experiment so the probability of one event will be the Complement of another event. For example – if A and B are two possible events, then
P(B) = 1 – P(A) or P(A’) = 1 – P(A).
P(A) + P(A′) = 1.
Conditional rule: When the probability of an event is given and the second is required for which first is given, then
P(B, given A) = P(A and B), P(A, given B). It can be vice versa
P(B∣A) = P(A∩B)/P(A)
Multiplication rule: Intersection of two other events i.e. events A and B need to occur simultaneously. Then
P(A and B) = P(A)⋅P(B).
P(A∩B) = P(A)⋅P(B∣A)
What is the probability of getting 4 heads on flipping a coin 12 times?
Solution:
Use the binomial distribution.
Lets suppose that the number of heads is r that represents the head times and in this case r = 4
Assuming that the coin is unbiased, you have a probability of success ‘p’(where p is considered as success) is 1/2 and the probability of failure ‘q’ is (where q is considered as failure).
The number of trials is represented by the ’n’ and here n = 12.
Now use the function for a binomial distribution:
P(R = r) = nCrprqn-r
P(R = 4) = (12C4)(1/2)4(1/2)12-4
= 495/4096
So the probability of flipping a coin 12 times and getting heads 4 times is 495/4096.
Similar Questions
Question 1: What are the chances of flipping 10 heads in a row?
Solution:
Probability of an event = (number of favorable event) / (total number of event).
P(B) = (occurrence of Event B) / (total number of event).
Probability of getting one head = 1/2.
here Tossing a coin is an independent event, its not dependent on how many times it has been tossed.
Probability of getting 2 heads in a row = probability of getting head first time × probability of getting head second time.
Probability of getting 2 head in a row = (1/2) × (1/2).
Therefore, the probability of getting 10 heads in a row = (1/2)10.
Question 2: What are the chances of flipping 12 heads in a row?
Solution:
Probability of an event = (number of favorable event) / (total number of event).
P(B) = (occurrence of Event B) / (total number of event).
Probability of getting one head = 1/2.
here Tossing a coin is an independent event, its not dependent on how many times it has been tossed.
Probability of getting 3 heads in a row = probability of getting head first time × probability of getting head second time x probability of getting head third time
Probability of getting 3 head in a row = (1/2) × (1/2) × (1/2)
Therefore, the probability of getting 20 heads in a row = (1/2)12
Question 3: If a coin is flipped 20 times, what is the probability of getting 6 heads?
Solution:
Use the binomial distribution.
Lets suppose that the number of heads is r that represents the head times and in this case r = 6
Assuming that the coin is unbiased, you have a probability of success ‘p’(where p is considered as success) is 1/2 and the probability of failure ‘q’ is (where q is considered as failure).
The number of trials is represented by the ’n’ and here n = 20.
Now use the function for a binomial distribution:
P(R = r) = nCr pr qn-r
P(R = 6) = (20C6)(1/2)6(1/2)12-6
= (20C6) (1/2)20
= 38760/1048576
= .0369644
So the probability of flipping a coin 20 times and getting heads 6 times is 0.0369644