Lecture 8

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What is the probability of a straight flush in Poker?

How many numbers are available for a given area code?

What is the probability of winning the Powerball lottery?

These and other types of problems can be calculated using various counting rules. Counting rules are useful where there are so many sample points that it would be impossible to list them individually.

Today we are going to look at some common counting rules

EX 1 - A student is deciding on a college and a major. She is considering five different colleges and three different majors. How many different distinct options are there.

This is an example of a problem where the multipicative rule is applicable

Multipicative Rule is used where you are selection one item from each of k distinct sets.

Number of choices = n₁n₂n₃…n_k

where n_k is the number of items in set k

What is k for the example problem?

How many options are there?

EX 1 - Four students is to be selected from a class of ten students. How many different combinations are there?   Note the order is important.

This is an example of permutation problem

Permutation Rule is used where you are selecting

n element from one set of N elements and are arranging the elements in a specific order.

Number of choices = P^N_n = N!/(N-n)!

What is N for the example problem?

What is n for the example problem?

How many permutations are there?

Ex: A class of 11 students is being divided into three groups. The size of the groups is 4, 4 and 3 respectively. How many possible distinct groups are there?

The is an example of the Partitions Rule problem

Partitions Rule is used to determine the number of partitions when a singe set of N items is being divided into k sets

Number of partitions = N!/(n1!n2!n3!…nk!)

note: n1+n2+n3….+nk = N

What is N for example?

Calculate number of partitions

Ex: Four students is to be selected from a class of ten students. How many different combinations are there? (Note the order in which the students are selected is not important.)

We can use the partitions rule.

What is value of k?

What is value of N?

What is value of n_ks?

 

 

This problem is a special application of partitions rule where k = 2.

We can also use the combination rule to calculate the result

Combination rule is used to calculate the number of different combinations possible when we are selecting n elements from a set of N elements.

In this case, the order is not important.

Number of different combination = N!/n!(N-n)!

Ex: Four students selected from ten

N? n?

Number of combinations?

What would order if we where interested in selecting the students in specific order?

Lecture 9

A jar contains 20 red marbles, 50 blue marbles, 30 white marbles. Your job is to select two marbles with replacement from this jar.

I. Use the appropriate counting rule to determine the number of sample points if we are interested in the color of the first and the color of the second marble selected.

II. Develop the theoretical probability distribution for the problem described in I. (State your assumptions.)

III Develop the theoretical probability distribution if we are interested in the number of blue balls selected.

IV Each person should randomly select two marbles from the jar. Record the results as a) the color of first and color of second, b) number of blue marbles.

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Date of last update - 23 Sep 1998