10 song combinations, equally likely. 90% chance of "Mama" song, due to more pairing possibilities.
Analyzing Renae's MP3 Player Playlist
a. Sample Space:
Since the order of songs doesn't matter, we can create the sample space by listing all unique combinations of two songs:
1. I Love My Mama - Don't Call Me Mama
2. I Love My Mama - Carefree and Blue
3. I Love My Mama - Go Back To Mama
4. I Love My Mama - Smashing Lollipops
5. Don't Call Me Mama - Carefree and Blue
6. Don't Call Me Mama - Go Back To Mama
7. Don't Call Me Mama - Smashing Lollipops
8. Carefree and Blue - Go Back To Mama
9. Carefree and Blue - Smashing Lollipops
10. Go Back To Mama - Smashing Lollipops
We can be sure we listed all combinations by systematically pairing each song with every other song once, ensuring no song is repeated or left out.
b. Equal Likelihood:
Yes, each combination of two songs is equally likely. This is because the MP3 player randomly selects songs without any bias. Each song has an equal chance of being chosen first, and then the second song is chosen with equal probability from the remaining songs.
This equal likelihood is crucial for calculating probabilities in the following parts.
c. Probability of "Mama" Songs:
There are 3 songs with "Mama" in the title: I Love My Mama, Don't Call Me Mama, and Go Back To Mama. We need to find the probability of selecting 2 songs where at least one has "Mama".
There are 3 ways to select one "Mama" song and 4 ways to choose the second song (any of the remaining songs). However, we've overcounted 3 combinations (each "Mama" song paired with itself). Therefore, the total favorable outcomes are (3 * 4) - 3 = 9.
The probability is then the number of favorable outcomes divided by the total number of outcomes (10): 9/10 = 0.9.
d. Probability of at least one "Mama" Song:
We can find this probability by considering the opposite event: the probability of NO "Mama" songs being selected. There's only 1 way to achieve this (picking Carefree and Blue with Smashing Lollipops).
Therefore, the probability of at least one "Mama" song is 1 minus the probability of no "Mama" songs: 1 - 1/10 = 9/10.
e. Higher Probability for at least one "Mama" Song:
The probability of at least one "Mama" song being higher than the probability of exactly two "Mama" songs makes sense because there are more ways for at least one "Mama" song to occur. There are 3 "Mama" songs, and each can be paired with any of the 4 non-"Mama" songs, resulting in 12 possible outcomes.
However, for exactly two "Mama" songs, we only have 3 ways to choose the first "Mama" song and then 2 ways to choose the second (since we can't pick the same one again). This results in only 6 possible outcomes.
Therefore, there are twice as many ways for at least one "Mama" song to occur compared to exactly two, justifying the higher probability in part (d).