Cd 13 tracks

Question

Suppose a CD you bought has 13 tracks. After listening to the CD you decide you like 5 of the songs. the random feature on your CD player will play eah of he 13 songs once in random order. Find the probability that among the first four songs played 

 

A. You like 2 of them

B. You like 3 of them

C. You like all of them

Answer 1

I'm assuming you're familiar with the binomial coefficient, n C k.

 

The total number of ways to choose 4 out of 13 songs is given by 13C4 = 715. This is the size of your sample space for all three parts.

 

A) You like 2 of them and don't like 2 of them. There are 5C2 = 10 ways to choose 2 of the 5 you like and 8C2=28 ways to choose 2 of the 8 you don't like. The total probability will be

10*28/715 = 280/715 = 56/143

 

B) Similarly, if you like 3 of them and don't like 1 of them, you get

5C3*8C1/13C4 = 10*8/715 = 80/715 = 16/143

 

C) You like all 5 of them:

5C4*8C0/13C4 = 5*1/715 = 5/715 = 1/143

Answer 2

 A.

 

Consider the first song played.  The probability of its being one you like is (5/13)

Suppose the second song played is also one that you like. The probability of this happening is (4/12)

Things change a bit when were considering the songs you don't like.  The probability of one of their being played is 8/13 and the probability of two of their played is (7/12)*(8/13)

 

Overall probability is (5/13)*(4/12)*(8/11)*(7/10) = (28/429)

 

B. 

 

Overall probability is (5/13)*(4/12)*(3/11)*(8/10) = (4/143)

 

C.  

 

Overall probability is (5/13)*(4/12)*(3/11)*(2/10) = (1/143)

 

It might be instructive at this point to consider the Answers to A. B. and C. over the common denominator 429.  (429 = 143*3)

 

A. = 28/429, B. = 12/429 and C. = 3/429.

 

Based on intuition alone, you would expect A to be likelier than B. to be likelier than C.  We have just shown this to be the case mathematically.