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 not sure about the outcome of an event, take help of the probabilities of certain outcomes, how likely they occur. For a proper understanding of probability, 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
There are different terms used in probability that are not commonly used normally, terms like experiments, sample space, favourable outcome, trial, random experiment, etc. Lets take a look at their definitions in details,
- 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 a coin is tossed, a head or tail is obtained but the outcome is not sure 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 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. One 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 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)
How to find the probability of flipping multiple coins?
Solution:
Coin flip probabilities only deal with events related to a single or multiple flips of a fair coin. A toss of fair coin has an equally likely chance of coming up Heads or Tails.
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. For example, whenever a coin is flipped, either heads (H) or tails (T) is obtained. So, here the sample space is {H, T}. Every subset of a sample space is called an event.
P(A) = Favorable outcomes / Total number of outcomes
Sample Problems
Question 1: What is the probability of flipping one coin?
Solution:
To calculate the probability of the event, It contains only one element and sample space contains two elements, so the sample space will be {H, T}
So, total number of outcome is 2.
What is the probability of a coin landing on tails or head?
The probability of landing on head is given as: P(A) = Favorable outcomes / Total number of outcomes
= 1/2
Same for the probability of landing on tail, P(A) = Favorable outcomes / Total number of outcomes
= 1/2
Question 2: What is the probability of flipping two coin?
Solution:
To calculate the probability of event, by flipping of two coins,
Then the sample space will be {HH, HT, TH, TT}
Total number of outcome = 4
Example: Find the probability of,
- At least two Heads.
- Atmost one Heads and on tail.
- One Tail
P(A) = Favorable outcomes / Total number of outcomes
- Probability of At least two Heads
Favorable outcomes of having two head = HH
= 1
Probability of having two Heads P(A) = Favorable outcomes / Total number of outcomes
= 1/4
- Probability of Atleast one tail and one head
Favorable outcomes of having one tail and one head = HT, TH
= 2
Therefore,
Probability of At least one Tail and one head P(A) = Favorable outcomes / Total number of outcomes
= 2/4
= 1/2
- Probability of getting one tail
Favorable outcomes of having one tail = HT, TH
= 2
Total number of outcome = 4
Therefore, probability of having two Tail P(A) = Favorable outcomes / Total number of outcomes
= 2/4
= 1/2
Question 3: What is the probability of flipping three coins? Find the probability of these events?
- Getting TWO Heads.
- Getting all Tails.
Solution:
To calculate the probability of event, by flipping of three coins
Then the sample space will be {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
Total number of outcome = 8
P(A) = Favorable outcomes / Total number of outcomes
So now,
- Probability of getting two heads
Sample space is {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
Here the favourable outcome of having Two heads = 3
So the probability of that event,
P(A) = Favorable outcomes / Total number of outcomes
= 3/8
- Probability of getting two Tails
Sample space is {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
Here the favourable outcome of having all tails = 3
So the probability of that event,
P(A) = Favorable outcomes / Total number of outcomes
= 3/8
Question 4: A coin is tossed 20 times. What is the probability of getting at least 1 tail?
Solution:
Use the binomial distribution,
Tossing a coin can give 2 outcomes.
So, tossing a coin 20 times can give (220) outcomes.
If the outcomes of getting at least one tail are excluded; we will be left with the one and only option of getting all ‘heads’.
So, the probability of getting at least one tail = [{(220) – 1}/(220)]
= [1 – {1 / (220)}].
= 0.999999
Question 5: What are the odds of flipping tails 10 times 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 a tails = 1/2.
Tossing a coin is an independent event, it is not dependent on how many times it’s been tossed.
Probability of getting 2 tails in a row = probability of getting tail first time × probability of getting tail second time.
Probability of getting 2 tails in a row = (1/2) × (1/2).
Similarly, the probability of getting 10 tails in a row = (1/2)12.