Suppose that a class contains 15 boys and 30 girls and that ten students are to be selected at random for a particular assignment. Find the probability that exactly 3 boys will be chosen.
The question above represents a scenario that may require the application of the combination rule. The combination rule involves the situation in which one finds the most appropriate way that they can select r objects from a population of n subjects following no definite order. Through this approach, one can utilize the given information and generate the provided components to help in making a decision.
In this case, the formula to use is nCr=n! / (n-r)! r!
The selection will involve 3 boys and 7 girls.
In that case, (15C3* 30C7)/ 45C10
The combination of 15 and 3 will give;
15C3=15!/ ((15-3)!3! = 455. This case represents the various ways that a researcher can successfully select 3 boys from a population of 15 boys
30C7=30! / ((30-7)!7!)=2035800. This representation offers the various approaches that one can obtain 7 girls from a population containing 30 girls.
45C10=45! / ((45-10)! 10!)= 3190187800. This equation represents the multiple ways that one can select 10 students (both boys and girls) from a population containing 45 students.
In the attempt to obtain 3 boys from a selection containing 10 students, one needs to look for the product of the combination of 7 girls and 3 boys. The researcher may then need to look for the quotient of the above mixture and the combination of total students.
In this case, 455*2035800=926289000
By division we get, 926289000/ 3190187800= 0.29030
Thus the probability of obtaining 3 boys from a selection of 10 students= 0.2903 or 29.03
Follow up question from the instructor:
Did you use a calculator with a nCr button or Excel?
Why do you calculate (15C3* 30C7)/ 45C10 ? Why do you multiply and divide these combinations?