Question

while waiting for a bus after school, renae programmed her mp3 player to randomly play two songs from her playlist, below. assume that the mp3 player will not play the same song twice.

playlist
a. i love my mama (country)
by the strings of heaven
b. don’t call me mama (country) duet
by sapphire and hank tumbleweed
c. carefree and blue (r & b)
by sapphire and prism escape
d. go back to mama (rock) duet
by bjorn free and sapphire
e. smashing lollipops (rock)
by sapphire

a sample space is a list of all possible outcomes for a probabilistic situation. list the sample space for all the combinations of two songs that renae could select. the order that she hears the songs does not matter for your list.

how can you be sure that you listed all of the song combinations?

you should find 10 different possible combinations. you will need this information for the next few problems. make sure you can find all 10 combinations on your own.

○ i found all ten combinations! i am confident i can do it again in the future.
○ 10? i have no idea how there are 10 combinations!

Explanation:

To determine the sample space of two - song combinations (where order doesn't matter) from a playlist with 5 songs (a, b, c, d, e), we use the combination concept. The formula for combinations of \(n\) items taken \(r\) at a time is \(C(n,r)=\frac{n!}{r!(n - r)!}\), here \(n = 5\) and \(r=2\), and \(C(5,2)=\frac{5!}{2!(5 - 2)!}=\frac{5\times4}{2\times1}=10\).

Step 1: Start with song a

We pair song a with each of the other songs.

• Pair (a, b)
• Pair (a, c)
• Pair (a, d)
• Pair (a, e)

Step 2: Start with song b (excluding the pair with a which we already did)

• Pair (b, c)
• Pair (b, d)
• Pair (b, e)

Step 3: Start with song c (excluding pairs with a and b which we already did)

• Pair (c, d)
• Pair (c, e)

Step 4: Start with song d (excluding pairs with a, b, c which we already did)

• Pair (d, e)

The sample space (all two - song combinations) is: \(\{(a,b),(a,c),(a,d),(a,e),(b,c),(b,d),(b,e),(c,d),(c,e),(d,e)\}\)

To be sure we have all combinations, we can use the combination formula \(C(n,r)\) (where \(n = 5\) songs and \(r = 2\) songs chosen at a time) to calculate the number of combinations. \(C(5,2)=\frac{5!}{2!(5 - 2)!}=\frac{5\times4\times3!}{2\times1\times3!}=10\), and by systematically pairing each song with the songs that come after it (to avoid duplicates due to non - ordered pairs), we can list all 10 combinations as we did above.

If we have found all ten combinations, we would select the option: "I found all ten combinations! I am confident I can do it again in the future."

Answer:

The sample space is \(\{(a,b),(a,c),(a,d),(a,e),(b,c),(b,d),(b,e),(c,d),(c,e),(d,e)\}\). If we found all 10 combinations, the answer to the multiple - choice part is: I found all ten combinations! I am confident I can do it again in the future.