How To Calculate Probability using Combination

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.

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 in Math

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:

ⁿC_r = 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.

What is Probability?

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,

Among the above mentioned formulas, the two most important are,

Combination with Repetition

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, ⁿC_r = (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.

Combination without Repetition

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, ⁿC_r = 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:

Understand the Mathematical Notation of Combination

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:

• Factorial notation: Factorial notation is a exclamatory mark right to the next of a number. It is a mathematical symbol. For example: 3! is equivalent to 3 × 2 × 1.
• Pascal’s triangle: We can find solutions of several combination probability problems by using Pascal’s triangle concept.
• Variable “n” : variable n represents the total number of outcomes of an event. For example, A lock with 3 digits, each of which can be 0 to 9, has an n value of 10 because there are 10 possible digits for each spot in combination.
• Variable “r” : variable r is used to identify the possible items of an event.

Identify the Style of Calculation

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.

Choose the Appropriate Formula

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 = n^r
• 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)!

Input variables and Calculate the Probability

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

• Baye’s Theorem
• Probability Distribution
• Conditional Probability

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?

Why do we use combination probability?

We use combination probability to calculate the possible outcome of an event where the order of outcome does not matter.

How is the probability of a combination is related to the probabilities of its components?

Here components refer to the individual events of a combination. The probability of a combination = The product of the probabilities of the individual components.

What is the formula of Combination?

ⁿC_r = n!/(n-r)!r!

where

• n = total number items,
• r = number of items to be chosen at a time.

Can combinations have repetition?

In permutation, order matters and repetition is not allowed whereas in combination, order of outcomes does not matter, but the repetition is allowed.

What are the three rules of probability?

The three rules of probability are:

• multiplication rule
• addition rule
• complement rule