Basics of permutations and combinations and how to use them in probability

Permutations and combinations are fundamental concepts in probability and statistics. In simple terms, permutations are the ways in which objects from a set can be arranged, while combinations are the ways in which objects from a set can be selected.

To understand these concepts better, let’s take an example of a deck of cards. If we want to know the number of permutations possible with a deck of cards, we need to consider the number of cards and the order in which they are arranged. For example, if we have 5 cards in a deck and we want to arrange them in a particular order, there will be 5! (5 x 4 x 3 x 2 x 1) possible permutations.

On the other hand, if we want to know the number of combinations possible with a deck of cards, we only need to consider the number of cards and not the order in which they are selected. For example, if we have 5 cards in a deck and we want to select 3 cards, there will be 5C3 (5!/3!2!) possible combinations.

Now let’s see how we can use permutations and combinations in probability.

Suppose we have a bag containing 3 red balls, 2 blue balls, and 1 green ball. If we randomly pick a ball from the bag without looking, the probability of picking a red ball is 3/6, a blue ball is 2/6, and a green ball is 1/6. This is an example of a simple probability where we can directly calculate the probability by dividing the number of desired outcomes by the total number of outcomes.

However, in some cases, we may need to use permutations and combinations to calculate the probability. For example, suppose we want to know the probability of picking 3 balls from the bag such that there is at least one red ball. In this case, we can calculate the probability by dividing the number of favourable outcomes by the total number of outcomes.