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Probabilities Involving Combinations

Example1: The grade appeal process at a university requires that a jury be structured by selecting five individuals randomly from a pool of eight students and ten faculty.

(a)  What is the probability of selecting a jury of all students? 

(b)  What is the probability of selecting a jury of all faculty?  

(c) What is the probability of selecting a jury of two students and three faculty?In this exercise, there are two groups made u of 8 students and 10 faculty. The combined number in the two groups is 18. Five individuals are to be selected from the total.

(a.) To select all students, you must choose 5 students from the group of 8 students and 0 faculty from the group of 10. Using the combination formula, you will do the following calculation: .

To calculate the numerator, enter the first value, 8 press MATH, highlight PRB and select 3:nCr and enter the next value, 5. Next press x and enter the next value, 10, press MATH, highlight PRB and select 3;nCr enter the next value,0. Next press and enter the first value in the denominator, 18, press MATH, highlight PRB and select 3:nCr, enter the next value, 5. Press ENTER and the answer will be displayed on your screen.

(b.) Repeat the steps in part (a.) but select 0 students and 5 faculty.

(c.) Repeat the steps in part (a.) but select 2 students and 3 faculty.

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Example 2: Suppose a compact disk (CD) you just purchased has 13 tracks.  After listening to the CD, you decide that you like 5 of the songs.  The random feature on your CD player will play each of the 13 songs once in a random order.  Find the probability that, among the first 4 songs played,

(a)  You like 2 of them. 

(b)  You like 3 of them.

(c)  You like all 4 of them.

The compact disk has a total of 13 songs. Of the 13 songs, there are 5 songs that you like and 8 songs that you do not like. Suppose that four songs are randomly selected and played.

(a.) Calculate the probability that, among these first four songs selected, you like exactly two of them. Using the combination formula, you will do the following calculation  

                     .

To calculate the numerator, enter the  first value, 5, press MATH, highlight PRB and select 3;nCr and enter the next value, 2. Next press x and enter the next value, 8, press MATH, highlight PRB and select 3;nCr, enter the next value, 2. Next press and enter the first value in the denominator, 13, press MATH, highlight PRB and select 3:nCr, enter the next value, 4. Press ENTER and the answer will be displayed on your screen.

(b.) Repeat the steps in part (a.) but select 3 songs from the group of songs that you like and 1 from the group of songs that you do not like.

(c.) Repeat the steps in part (a.) but select 4 songs from the group of songs that you like and 0 songs from the group that you do not like.