ork 3 (Deadline: Feb 7) here 10. Suppose that fifteen new students are evenly di

Question

ork 3 (Deadline: Feb 7) here 10. Suppose that fifteen new students are evenly distributed among three classes and there are three whiz-kids among the fifteen. What is the probability that each class gets one whiz-kid? What is the probability that one class gets all the three whiz-kids? 11. If six dice are rolled, what is the probability that five dice show the same number and dice shows a different number? 12. If six dice are rolled, what is the probability of getting a quadruple and a pair? For example, (1, 1,1,1,2,2) has a quadruple and a pair 13. If six dice are rilled, what is the probability of getting two triples? For example, (1, 1, 1,2,2,2) has two triples.

Explanation / Answer

10.

These 15 students are evenly distributed among 3 classes, i.e. each class will have 15/3 = 5 students.

This is partition problem, partioning 15 students in 5, 5, 5 (3 classes)

so, total number of outcomes would be 15!/(5!5!5!) = 3632428800 (all possible ways)

Now, to get probability to get whiz kid in each class, first assign one whiz kid in each class, this can be done in 3! ways and then remaining 4 students from (15 - 3) = 12 students

so, 3! * 12!/(4!4!4!) = 39916800 (possible ways to get whiz kid in each class)

Probability that each class gets one whiz kid is 39916800/3632428800 = 0.01099

Next, one class gets all whiz kid will be in 3 ways (class 1 gets all three whiz kid or class 2 gets all three or class 3 gets all three)

in remaining 12 students, 2 will go in the class which gets all 3 whiz kids, and other two will get 5 each

3 * 12!/(2!5!5!) = 2993760

So, probability that one class gets all three whiz-kid is 2993760/3632428800 = 0.0008

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