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VOOZH | about |
Combination probability is a mathematical method that involves the process of combination in determining the number of favorable outcomes of an event. We use combinations in probability problems to determine a sequence of outcomes where the order of the outcomes does not matter. Understanding how to calculate combination probability can be a useful mathematical skill in the field of math and science.
In this article, we will discuss in detail the definitions of combination and probability and how to calculate combination probability with solved examples.
Table of Content
Combination probability, also known as the probability of combinations, involves the process of determining the possibility of specific subset from a large set regardless to the order in which the subsets are chosen. Combination probability has wide range of application in games, statistics, gambling, genetics, etc.
Combination is known as the selection of items from a large set, where the order of the items does not matter. for example, if there is a set of n items from which we have to select r items, the formula to calculate the combination is:
nCr = n!/(n-r)!r!
where
- n = total number of items,
- r = number of items to be chosen at a time.
For example, let A, B, C are three components i.e., n = 3 and combinations size is 2, means r = 2. Then there are 3C2 such combinations present, which is equal to 3. The combinations are AB, BC, and CA. In probability problems, combinations are used to determine a sequence of outcomes where the order of the outcomes is not important.
Probability is the measure of the possibility that an event will occur. It is defined as a number between 0 and 1 with 0 denoting improbability and 1 denoting certainty. It is a branch of mathematics that deals with the interpretation of random events. For example, when we toss a coin, either we get head or tail, hence only two possible outcomes are possible, the outcomes are (H,T).
Formula for Probability:
Probability of a event to happen P(E) = Number of favorable outcomes/Total Number of outcomes
To calculate combination probability, it is important to use the correct formula to find the probability of a specific outcome. There are four formula's to calculate the probability. They are,
Formula | Equation |
|---|---|
Combination without repetition | n!/(n-r)!r! |
Combination with repetition | (r+n-1)!/r!(n-1)! |
Permutation with repetition | nr |
Permutation without repetition | n!/(n-r)! |
Among the above mentioned formulas, the two most important are,
When repetition is allowed, you can use the following equation to find out the number of combinations,
(r+n-1)!/r!(n-1)!
where,
- n = total number of outcomes of a set
- r = total possible number of outcomes at a time.
Example: Total number of balls in the pool= n = 5. The number of balls to be selected = r = 4, where the selection of balls can be repeated. The order of selection does not matter. Find the number ways the ball can be selected?
Solution:
We will use the formula, nCr = (r+n-1)!/r!(n-1)!
Now, putting the values we get, 5C4 = (4+5-1)! / 4!·(5-1)!
= 8! / 4!·4! = 8×7×6×5 / 4×3×2×1 = 70
∴ There are 70 possible ways to select the ball.
When repetition is not allowed, you can use the following equation to find out the number of combinations,
n!/(n-r)!r!
where,
- n = total number of outcomes of a set
- r = total possible number of outcomes at a time
Example: The principle would like to assemble a committee of 6 students from the 11 member student council. How many different committees can be chosen?
Solution:
We will use the formula, where, nCr = n!/(n-r)!r!
Now, after putting thee values we get, 11C6 = 11!/(11-6)!·6!
= 11! / 5!·6! = 11×10×9×8×7 / 5×4×3×2×1 =66×7 = 462
∴ 462 committees can be chosen.
Combination is a mathematical concept which helps us in determining the possible outcomes of an event. To calculate combination probability we have to follow several steps and utilize several formulas. They are discussed below:
In combination probability, different formulas are required to solve various problems. But the basic concept behind those formulas are same. To calculate combination probability, we use several terms and notations, which are:
In order to apply correct formula for calculating problems on combination, it is important to understand the type of calculation you are performing. The first thing to notice is whether the calculation is a permutation or combination. Then you must note that is there any repetition of the same value during the combination. At the end there will be four calculation types, one pair consisting repeat and no repeat of combination and another for permutation.
After identifying the correct style of the calculation, it is important to use the correct formula to find the probability of a specific outcome. The four combination probability equations are:
- Permutation with repetition , total permutation = nr
- Permutation without repetition, total permutation = n!/(n-r)!
- Combination without repetition , total combination = n!/(n-r)!r!
- Combination with repetition , total combination = (r+n-1)!/r!(n-1)!
After selecting appropriate formula, now put all the values to calculate the combination. The probability for each combination can be calculated by dividing the number of favorable combinations by total number of combinations. Then combining the number of favorable combinations with the individual probabilities, we get overall combination probability.
Also, Check
Example 1. In a team, 3 boys and 4 girls are there. Among them 4 members need to be selected for one round of a game. Find the probability of selecting an equal number of boys and girls?
Solution:
Probability of selecting an equal number of boys and girls = (4C2 × 3C2)/7C4
(4C2 × 3C2)/7C4 = 18/35
Hence probability of selecting equal number of boys and girls is 18/35
Example 2. What is the number of possible combinations to choose 6 numbers from a set of 49 numbers?
Solution:
49C6 = 49!/6! (49-6)!
49C6 = 49!/(6! × 43!) possible ways.
Example 3. What is the number of ways need to form a group of 3 people from a group of 10?
Solution:
10C3 = 10!/3!(10-3)!
10C3 = 10×9×8/3×2×1
10C3 = 120
∴There are 120 different ways to do this.
Example 4. How many ways can 8 students be chosen from a class of 21?
Solution:
21C8 = 21!/8!×(21-8)!
21C8 = (21×20×19×18×17×16×15×14)/(8×7×6×5×4×3×2×1)
21C8 = 203,490
∴ There are 203,490 ways to chose from a class of 21.
Q1. A committee of 4 people is to be selected from a group of 10 people. What is the probability that a specific person, Alice, is on the committee?
Q2. From a standard deck of 52 cards, 5 cards are drawn at random. What is the probability that all 5 cards are of the same suit?
Q3. A class has 12 boys and 8 girls. If a group of 5 students is selected at random, what is the probability that the group will consist of 3 boys and 2 girls?
Q4. A jar contains 7 red marbles and 5 blue marbles. If you draw 3 marbles at random, what is the probability that exactly 2 of them are red?
Q5. From a shelf of 8 math books and 6 science books, you randomly select 4 books. What is the probability that you select 2 math books and 2 science books?
