Probability is a mathematical branch that deals with calculating the likelihood of occurrence of a random event. Its value ranges between 0 (event will never occur) and 1 (event will certainly occur). Higher the value higher the chances of the event occurring. To determine the likelihood of an event first calculate the total number of possible outcomes and a total number of preferred outcomes. A simple example is the tossing of a fair (unbiased) dice. Since the dice are fair, the six outcomes (“1”, “2”, “3”, “4”, “5”, and “6”) are all equally probable and since no other outcomes are possible, the probability of either event is 1/6.
Terms in probability
There are certain important terms used in probability like sample space, outcome, probable events, impossible events, experiments, etc. Let’s learn about these terms in detail,
- Experiment: An activity that has a well-defined set of outcomes is an experiment. Each experiment has few favorable outcomes and some unfavorable outcomes combining to form sample space.
- Sample Space: All the outcomes of all the trials of an experiment. For example when a dice is rolled, the outcomes can be “1”, “2”, “3”, “4”, “5” and “6”. This will make our Sample Space. S = (“1”, “2”, “3”, “4”, “5” and “6”).
- Outcome: Possible results of an outcome. Every outcome of an experiment is one of its type i.e. unique. Only one outcome occurs for a trial of the experiment.
- Probable Event: Events that can be predicted. For example, calculate chances of rain for tomorrow, hence it is a probable event.
- Impossible Event: Event which has zero probability to occur, Or an event that is not a part of our experiment. For example when a fair dice is thrown the probability of occurrence of 7 is zero, hence it is an impossible event.
What are all the possible outcomes if a coin is tossed?
Answer:
Single Coin is Tossed
When a fair coin is tossed then there are two possible outcomes: H(head), T(tail). This probability of occurrence of both events will be 0.5.
When a coin that had been influenced is tossed then the possible outcomes can be different. For example, there is a doubly headed coin (i.e. both sides of coins are head), then there is only one possible outcome: H(head). In this case, the probability of occurrence of the head will be 1(certain).
Multiple coins are tossed
When n number of coins are tossed together then a total number of possible outcomes will be 2n. When 2 coins are tossed together, total possible outcomes = 22 = 4.
| H | H |
| H | T |
| T | H |
| T | T |
When 3 coins are tossed together, total possible outcomes = 23 = 8.
| H | H | H |
| H | H | T |
| H | T | H |
| H | T | T |
| T | H | H |
| T | H | T |
| T | T | H |
| T | T | T |
Trick
When writing outcomes of n simultaneously tossed coins,
- In the first column write 2n-1 times H (head) then write 2n-1 times T (tail).
- In the second column, write 2n-2 times H then write 2n-2 times T and repeat this step until the column is filled (2 times).
- For the third column, write 2n-3 times H then write 2n-3 times T and repeat 22 times.
- Continue this process until the nth coin is reached.
- For the nth coin write 1 time H then 1 time T and repeat for 2n-1 times.
Sample Problems
Question 1: What are the possible outcomes when a die is thrown?
Answer:
When a die is thrown possible outcomes will be (“1”, “2”, “3”, “4”, “5” and “6”). And there will be an equal probability (1/6) for each of them to occur.
Question 2: Given 6 red balls and 7 green balls. What are the possible outcomes of choosing one ball?
Answer:
In this case, there are 2 possible outcomes “red ball” and “green ball”. But the probability of both of them is different.
Total outcomes = 7 + 6 = 13. (red balls + green balls)
Probability of choosing a green ball will be = 7/13.
The probability of choosing a red ball will be = 6/13.
Question 3: What are possible outcomes when 2 dice are thrown simultaneously?
Answer:
When 2 dice are thrown simultaneously total possible outcomes will be = 62 = 36.
(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)
(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)
(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)
(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6)
(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)
(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)