Certainly! Let's break down the calculations for each part of the question step-by-step.
### Part (a): Probability that the first GG song heard is the fifth song played
1. Total number of songs: There are 100 songs in total, and 10 of them are by The Giggling Giraffes (GG), which means there are 90 non-GG songs.
2. Condition for the first four songs: The first four songs must be non-GG songs. The probability of selecting a non-GG song in each step, considering that songs are not repeated, can be calculated step-by-step:
- Probability that the first song is a non-GG song:
[tex]\[
\frac{90}{100} = 0.9
\][/tex]
- Probability that the second song is a non-GG song:
[tex]\[
\frac{89}{99} \approx 0.8989899
\][/tex]
- Probability that the third song is a non-GG song:
[tex]\[
\frac{88}{98} \approx 0.8979592
\][/tex]
- Probability that the fourth song is a non-GG song:
[tex]\[
\frac{87}{97} \approx 0.8969072
\][/tex]
3. Probability of the fifth song being a GG song: Since there have been four non-GG songs already played, the probability that the fifth song is a GG song is:
[tex]\[
\frac{10}{96} \approx 0.1041667
\][/tex]
4. Combined Probability: The combined probability that the first GG song heard is the fifth song played can be found by multiplying the probabilities:
[tex]\[
(0.9) \times (0.8989899) \times (0.8979592) \times (0.8969072) \times (0.1041667) \approx 0.0678782
\][/tex]
So, the probability that the first GG song heard is the fifth song played is approximately 0.0679 or 6.79%.
### Part (b): Probability that at least one of the first five songs played is by the GG
1. Complementary Probability: We will use the complement rule here. The complement of "at least one GG song in the first five songs" is "none of the first five songs are GG songs."
2. Calculate the probability that none of the first five songs are GG songs: This can be calculated step-by-step:
- Probability that the first song is a non-GG song:
[tex]\[
\frac{90}{100} = 0.9
\][/tex]
- Probability that the second song is a non-GG song:
[tex]\[
\frac{89}{99} \approx 0.8989899
\][/tex]
- Probability that the third song is a non-GG song:
[tex]\[
\frac{88}{98} \approx 0.8979592
\][/tex]
- Probability that the fourth song is a non-GG song:
[tex]\[
\frac{87}{97} \approx 0.8969072
\][/tex]
- Probability that the fifth song is a non-GG song:
[tex]\[
\frac{86}{96} \approx 0.8958333
\][/tex]
3. Combined Probability: Multiplying these probabilities:
[tex]\[
0.9 \times 0.8989899 \times 0.8979592 \times 0.8969072 \times 0.8958333 \approx 0.5837524
\][/tex]
4. Complementary Probability: The probability that at least one of the first five songs is by GG can be found by subtracting the above result from 1:
[tex]\[
1 - 0.5837524 \approx 0.4162476
\][/tex]
So, the probability that at least one of the first five songs played is by The Giggling Giraffes is approximately 0.4162 or 41.62%.
