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Probability is a mathematical theory that deals with the chances of a predictable outcome. It can also be defined as the analysis of phenomena that occurs randomly. For an event, there could be several outcomes, predicting a random outcome is not a probability. But instead, predicting a random outcome with favourable conditions is known as probability.
The probability of an event is predicted with a chance. The probability of a chance from an event lies between 0 to 1. For example, in the event of tossing a coin with heads and tails, the chances of getting heads are ½ .
Fundamentals of probability
Sample space: It is the set of all the possible outcomes that could occur from an event.
Eg: rolling dice has 6 outcomes, so the sample space is 6.
Union: For events A and B, the probability of getting multiple outcomes but no common outcomes. It is denoted by U.
Intersection: For an event, all the possible outcomes are called intersections in a sample space. It is denoted by ∩.
The intersection of A and B is defined as the set of all outcomes belonging to both the sets of A and B and is denoted A ∩ B; Union of the set is defined as all the outcomes which belong to set A or set B or both. It is denoted A U B.
Unconditional probability: It is defined as the occurrence of a particular outcome in an event with several outcomes. For example, in a class of 50 girls and 20 boys, the probability of choosing a girl is 50/70 = 0.714. It is denoted by P(A).
Conditional probability: It is defined as the possibility of an outcome in an event based on another event. In this concept, two events occur simultaneously and the events don’t necessarily have to be dependent on each other.
Theories of probability
Classical theory:
Classical theory is the simple application of the occurrence of an event. The outcome must be equally likely to occur. This is also called unconditional probability.
P(A) = s ⁄ n=number of ways event A can occur ⁄ Total number of all outcomes
Example :
The probability of getting six in a pack of 52 cards is?
Solution: In a set of cards, there are 4 suits of 13 cards each: Spades (black), Clubs (black), Hearts (red), and Diamonds (red). Each suite contains the following: numbers 2 through 10, king, queen, jack, and Ace. Therefore, one set of cards contains four 6s.
𝑃(A)= 4 ⁄ 52 =1 ⁄ 13=0.0769
Empirical probability formula:
This is used when actual data is given. In this formula, it is important to determine the total number of outcomes. The total number of outcomes can be done by using a table or can be determined using factorials. It is also called experimental probability. It can be defined as the number of outcomes of a chance in a particular event which is dependent on the probability of the outcome recurring.
P(A) = the number of times event A occurs ⁄ the number of times the experiment performed
Example: The probability of a couple having 3 girls and 1 boy in a family can be determined by :
The total outcomes is= 24 (the possibility is 2 because the child can either be a boy or a girl, and 4 is the total number of final outcomes)
Total outcome = 16.
Out of the 16 combinations, only 4 outcomes predict 3 girls and 1 boy;
Therefore P(A)= 4 ⁄ 16 =1 ⁄ 4=0.25
Factorial formula:
Sometimes, it is difficult to calculate the possible outcomes in the total outcomes. In those cases, the factorial formula is used. It is especially used when arranging all the variables within the same places.
Example:
Determine the number of ways 6 persons can sit in 6 empty seats at a cinema hall.
To determine the total number of ways 6 people can sit in 6 empty seats at a cinema hall, one person at a time should be filled. The first seat would be open to all 6 people. After one person sits down, there are only 5 more people left to sit in the next open seat. This goes on till only when one seat and one person are left out of the 6 people and seats. In other words, this can be represented as “6!” (6 factorial) and is defined as:
6 × 5 × 4 × 3 × 2 × 1 = 720
The total number of ways 6 people can sit in a 6 seat empty theatre is 720.
Permutations and combinations
Permutations are used when no replacements are allowed and one outcome can occur only one time. It is also used when the order in which the outcome occurs is important. Eg: while entering a password, it is important not to change the order of the password.
It is determined by:
nPr = n! ⁄ (n-r)!
The “n” is the total number of items, and the “r” is the number of ways they are being selected.
Example: The student body of a school is electing the President, Vice-President, Treasurer, and Secretary from a batch of 14 students consisting of 5 seniors, 4 juniors, 3 10th graders, and 2 9th graders. Determine the probability of electing all seniors.
In this question, the order in which the students are elected matters, therefore, permutations are used.
14P4= 14! ⁄ (14-4)!= 24024
The next step is to figure out how many of the 24,024 arrangements are exclusively for seniors.
As a result, we’d like to see how many different ways we can arrange solely the five seniors as class representatives.
5P4 = 5! ⁄ (5-4)!=120
Therefore
P = 12024024= 0.005
Combinations are used when replacements are allowed and the order in which the sequence occurs does not matter. Eg: selecting toppings for a pizza, the order in which the topping is placed does not matter.
nCr= n! ⁄ (n-r)!r!
Example: In order to win the Slot 5 lottery, a player must exactly select the correct 5 numbers in any order from between 1 and 52. Determine the probability of winning the Slot 5 lottery.
Solution: Since the order of the numbers does not matter, we can use combinations to determine the total possible number of combinations of 5 numbers from 1-52.
52C5= 52! ⁄ (52-5)!5!= 2598960
It is known that, out of all the 2598960 possibilities only one outcome would determine the correct 5 numbers, the probability of winning the Slot 5 lottery is:1 ⁄ 2598960 = 0.000000385
Addition rule
This is used when the outcome involves two events occurring at the same time. P (A or B) = P (occurring of event A or B or both at the same time)
The formula is given by:
P (A or B)= P(A) + P(B)- P(A and B)
P(A and B) is subtracted because it involves outcomes from both A and B which means, each outcome would be accounted for twice.
If an event A and an event B are cannot happen at the same time, the formula simplifies to
𝑃(𝐴 𝑜r 𝐵) = 𝑃(𝐴) + 𝑃(𝐵)
Example: A circular spinner has 4 equal sectors coloured blue, yellow, orange and light green. What is the probability of landing on yellow or blue after spinning this spinner?
P(Yellow)= 1 ⁄ 4
P(blue)= ¼
P(red or blue)= ¼ + ¼
= 2/4
=½
Conditional probability
The conditional probability is the likelihood that event B will occur after event A. P (B|A) denotes the conditional probability. When the “and” in P(A and B) refers to the likelihood of event A occurring before event B, not when the “and” refers to the probability of both events A and B occurring at the same time, conditional probability is utilised.
Without replacement, conditional probability occurs. Dependent and independent conditional probabilities are the two types of conditional probabilities.
For dependent events, the following conditional formula is used: 𝑃 (𝐴 and 𝐵) = 𝑃(𝐴) · 𝑃 (𝐵|𝐴)
The following conditional formula is used for the independent event: P(A and B)= P(A). P (B)
Conclusion
Therefore, it is clear that probability theory is the chance of a possible outcome in an event occurring. Probability is commonly stated as the ratio of the number of possible outcomes to the actual number of outcomes in a given event. Different formulas are used in different situations of an event.
