A branch of mathematics that deals with the happening of a random event is termed probability. 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.
Probability
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 the outcome of an event is not sure, 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. It is already known 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 = Favorable outcomes / Total number of outcomes
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 two dice are rolled 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 the outcome is not confirmed 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 a coin is tossed, there are equal chances of getting ahead 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. The same weather again and again can not be experienced.
- 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 Formulae
- 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 a head in a fair coin toss?
Solution:
If a fair coin is tossed then the sample space will be {H, T}
Therefore total number of event = 2
Probability of having head = 1
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.
Similar Problems
Question1: What are the chances of flipping 5 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 5 heads in a row = (1/2)5.
Question 2: What are the chances of flipping 20 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 × 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)20
Question 3: What are the chances of flipping 10 tails 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 tail = 1/2.
Here, if Tossing a coin is an independent event, its not dependent on how many times it has been tossed.
Probability of getting 3 tails in a row = probability of getting tail first time × probability of getting tail second time × probability of getting tail third time
Probability of getting 3 tails in a row = (1/2) × (1/2) × (1/2)
Therefore, the probability of getting 10 tails in a row = (1/2)10
Question 4: When a fair coin is tossed what is the probability of getting a tail?
Solution:
If a fair coin is tossed then the sample space will be {H, T}
Therefore total number of event = 2
Probability of having tail = 1
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 tail = 1/2.
Question 5: What are the chances of flipping 20 tails 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 tail = 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 tails in a row = probability of getting tail first time × probability of getting tail second time × probability of getting tail third time
Probability of getting 3 tails in a row = (1/2) × (1/2) × (1/2)
Therefore, the probability of getting 20 tails in a row = (1/2)20