The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of favorable outcomes and the total number of outcomes.
Sometimes students get mistaken for “favorable outcome” with “desirable outcome”. This is the basic formula. But there are some more formulas for different situations or events.
1) There are 6 pillows in a bed, 3 are red, 2 are yellow and 1 is blue. What is the probability of picking a yellow pillow?
Ans: The probability is equal to the number of yellow pillows in the bed divided by the total number of pillows, i.e.
2) There is a container full of colored bottles, red, blue, green and orange. Some of the bottles are picked out and displaced. Sumit did this 1000 times and got the following results:
No. of blue bottles picked out:
No. of red bottles:
No. of green bottles:
No. of orange bottles:
a) What is the probability that Sumit will pick a green bottle?
Ans: For every bottles picked out, are green.
Therefore,
b) If there are 100 bottles in the container, how many of them are likely to be green?
Ans: The experiment implies that out of bottles are green.
Therefore, out of bottles, are green.
The tree diagram helps to organize and visualize the different possible outcomes. Branches and ends of the tree are two main positions. Probability of each branch is written on the branch, whereas the ends are containing the final outcome. Tree diagram used to figure out when to multiply and when to add. You can see below a tree diagram for the coin:
There are three major types of probabilities:
Theoretical Probability
Experimental Probability
Axiomatic Probability
It is based on the possible chances of something to happen. The theoretical probability is mainly based on the reasoning behind probability. For example, if a coin is tossed, the theoretical probability of getting head will be
It is based on the basis of the observations of an experiment. The experimental probability can be calculated based on the number of possible outcomes by the total number of trials. For example, if a coin is tossed 10 times and heads is recorded 6 times then, the experimental probability for heads is .
In axiomatic probability, a set of rules or axioms are set which applies to all types. These axioms are set by Kolmogorov and are known as Kolmogorov’s three axioms. With the axiomatic approach to probability, the chances of occurrence or non-occurrence of the events can be quantified. The axiomatic probability lesson covers this concept in detail with Kolmogorov’s three rules (axioms) along with various examples.
Conditional Probability is the likelihood of an event or outcome occurring based on the occurrence of a previous event or outcome.
Assume an event E can occur in r ways out of a sum of n probable or possible equally likely ways. Then the probability of happening of the event or its success is expressed as;
The probability that the event will not occur or known as its failure is expressed as:
E’ represents that the event will not occur.
Therefore, now we can say,
This means that the total of all the probabilities in any random test or experiment is equal to 1.
When the events have the same theoretical probability of happening, then they are called equally likely events. The results of a sample space are called equally likely if all of them have the same probability of occurring. For example, if you throw a die, then the probability of getting 1 is Similarly, the probability of getting all the numbers from one at a time is Hence, the following are some examples of equally likely events when throwing a die:
Getting 3 and 5 on throwing a die
Getting an even number and an odd number on a die
Getting 1, 2 or 3 on rolling a die
are equally likely events, since the probabilities of each event are equal.
The possibility that there will be only two outcomes which states that an event will occur or not. Like a person will come or not come to your house, getting a job or not getting a job, etc. are examples of complementary events. Basically, the complement of an event occurring in the exact opposite that the probability of it is not occurring. Some more examples are:
It will rain or not rain today
The student will pass the exam or not pass.
You win the lottery, or you don’t.
Also, read:
Independent Events
Mutually Exclusive Events
The Probability Density Function (PDF) is the probability function which is represented for the density of a continuous random variable lying between a certain range of values. Probability Density Function explains the normal distribution and how mean and deviation exists. The standard normal distribution is used to create a database or statistics, which are often used in science to represent the real-valued variables, whose distribution are not known.
Some of the important probability terms are discussed here:
Term | Definition | Example |
Sample Space | The set of all the possible outcomes to occur in any trial | 1. Tossing a coin, Sample Space 2. Rolling a die, Sample Space |
Sample Point | It is one of the possible results | In a deck of Cards: 4 of hearts is a sample point. the queen of Clubs is a sample point. |
Experiment or Trial | A series of actions where the outcomes are always uncertain. | The tossing of a coin, Selecting a card from a deck of cards, throwing a dice. |
Event | It is a single outcome of an experiment. | Getting a Heads while tossing a coin is an event. |
Outcome | Possible result of a trial/experiment | T (tail) is a possible outcome when a coin is tossed. |
Complimentary event | The non-happening events. The complement of an event A is the event not A (or A’) | Standard 52-card deck, Draw a heart, then Don’t draw a heart |
Impossible Event | The event cannot happen | In tossing a coin, impossible to get both head and tail |
Question 1: Find the probability of rolling a ‘3 with a die.’
Solution:
Sample Space
Number of favorable events = 1
Total number of outcomes = 6
Formula,
Put the value,
Thus, Probability,
Question 2: Draw a random card from a pack of cards. What is the probability that the card drawn is a face card?
Solution:
A standard deck has .
Total number of outcomes
Number of favorable events (considered Jack, Queen and King only)
Probability,
Question 3: A vessel contains 4 blue balls, 5 red balls and 11 white balls. If three balls are drawn from the vessel at random, what is the probability that the first ball is red, the second ball is blue, and the third ball is white?
Solution: The probability to get first ball is red or the first event is .
Now, since we have drawn a ball for the first event to occur, then the number of possibilities left for the second event to occur is
Hence, the probability of getting second ball as blue or the second event is
Again, with the first and second event occurred, the number of possibilities left for the third event to occur is
And the probability of the third ball is white, or third event is
Therefore, the probability is
Or we can express it as
Question 4: Two dice are rolled, find the probability that the sum is:
equal to 1
equal to 4
less than 13
Solution:
1) To find the probability that the sum is equal to 1 we have to first determine the sample space S of two dice as shown below.
1) Let E be the event “sum equal to 1”. Since, there are no outcomes which where a sum is equal to 1, hence,
2) Three possible outcomes give a sum equal to 4 such as;
Hence,
3) From the sample space, we can see all possible outcomes, , give a sum less than 13. Like:
So, you can see the limit of an event to occur is when both dies have number 6, i.e. Hence,
Example: 5 In Class, 30% of the students offered English, 20% offered Hindi and 10% offered both. If a student is selected at random, what is the probability that he has offered English or Hindi?
Solution:
Probability is a branch of mathematics that deals with occurrence of a random event. For example, when a coin is tossed in the air, the possible outcomes are Head and Tail.
The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of outcomes and the total number of outcomes.
There are three major types of probabilities: Theoretical Probability, Experimental Probability, Axiomatic Probability
If A and B are two events, then,