Probability: Permutation, Combination, and Their Role in Probability

Probability: Understanding Permutation and Combination

When discussing probability, the concepts of permutation and combination are essential. These two methods allow us to count the number of outcomes when selecting items without replacement or considering the order of selection. In this article, we will explore these concepts and how they relate to probability.

Permutation

A permutation is a specific arrangement of items from a larger set. It involves ordering the chosen elements and considering their positions. Suppose we have a set of n items, denoted as S = {s₀, s₁, ..., sn₋₂}, and we want to find all the possible arrangements of r items, where 0 < r <= n. The formula to compute the number of permutations (denoted as P(n,r)) is given by:

P(n,r) = n! / (n-r)!

Here, "!" denotes the factorial of a number, and factorials are defined recursively as follows:

n! = n * (n-1)!, when n > 0
0! = 1

For example, if we have a set S = {1, 2, 3} and we want to find all the possible arrangements of two items, we would have:

P(3,2) = 3! / (3-2)! = 3! / 1! = 3 × 2 × 1 = 6

In this case, the six permutations would be: 12, 13, 21, 23, 31, 32.

Permutations are useful when the order of the elements matters, such as in scenarios related to racing events, elections, or any situation where sequencing plays a role.

Combination

On the other hand, a combination is a selection of items from a larger set without regard to their position or order of arrangement. Given a set S = {s₀, s₁, ..., sn₋₂} and a positive integer r, the number of combinations of size r from S is denoted as C(n,r). The formula to compute the number of combinations (denoted as C(n,r)) is given by:

C(n,r) = nCr = n! / (n-r)! * r!

Again, "!" denotes the factorial of a number. For example, if we have a set S = {1, 2, 3} and we want to find all the possible selections of two items, we would have:

C(3,2) = 3! / (3-2)! * 2! = 3! / 1! * 2! = 3 × 2 × 1 = 6

In this case, the six combinations would be: 12, 13, 21, 23, 31, 32.

Combinations are useful when the order of the elements does not matter, such as in scenarios related to serial numbers, team selection, or any situation where sequencing is not a concern.

Probability

When considering probability, permutations and combinations come into play when dealing with various scenarios involving randomness.

Consider an experiment with n possible outcomes, each outcome being equally likely. The theoretical probability p of an event occurring is simply the ratio of the favorable outcomes to the total possible outcomes:

p(event occurring) = favorable outcomes / total possible outcomes

If we know the total number of outcomes and the number of favorable outcomes, we can determine the probability of the event.

Now, let's consider an example. Suppose we draw cards randomly from an ordinary deck (with 52 cards), and we want to find the probability of drawing a heart or a face card during the draw. In this scenario, there are 13 hearts in the deck and 12 face cards, making a total of 25 favorable outcomes. However, since each card can be drawn twice (once for each draw), there are actually 25 × 2 = 50 favorable outcomes. Since there are 52 cards in total, the probability of drawing a heart or a face card is:

p(drawing a heart or a face card) = 50 / 52

Alternatively, we can express this probability as a fraction or decimal:

p(drawing a heart or a face card) = 25/52 = 0.48077

This probability represents the likelihood of drawing a heart or a face card from a single draw.