### Counting Techniques for Probability Ratio using Combination

The probability ratio of an event is the likelihood of the chance that the event will occur as a result of a random experiment, and it can be found using combination. When the number of possible outcomes of a random experiment is infinite, the enumeration or counting of the sample space becomes tedious. In these situation we make use of what is known the counting technique principle for finding probability. Combinations will be used if the order of the events are not significant.

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$$ \binom{n}{k} = \dfrac{n!}{(n-k)!k!} $$

Let’s take a look at the worked examples now.

#### Probability using Combination

A bag contains \(n\) pink cards, \(n\) yellow cards and \(n\) green cards. Three cards are drawn at random from the bag, one at a time, without replacement.

(a) Find the probability, \(P_s \), that the three cards are all the same colour.

(b) Find the probability, \(P_s \), that the three cards are all the different colour.

(c) Find the probability, \(P_m \), that the two cards are of one colour and the third is of a different colour.

(d) If \(n\) is large, find the approximate probability ratio \(P_s : P_d : P_m \).