A particular iPod playlist contains 100 songs, 10 of which are by the Beatles. Suppose the shuffle feature is used to play the songs in random order with no repeating. What is the probability that the first Beatles song heard is the fifth song played? What is the probability that there are exactly two Beatles songs in the first five songs played?

1. The 1st Beatles song heard is the 5th song played means that first 4 songs are not Beatles songs (there are 100 - 10 = 90 such songs) and the 5th song is Beatles song (there are 10 such songs). Hence, the probability that the first Beatles song heard is the fifth song played is

[tex]\dfrac{C^{90}_4\cdot C^{10}_1}{C^{100}_5}=\dfrac{2,555,190\cdot 10}{75,287,520}\approx 0.3394[/tex]

2. There are exactly two Beatles songs in the first five songs played means there are 2 Beatles songs and 3 not Betles songs (the number when these songs are played do not matter). Hence, the probability that there are exactly two Beatles songs in the first five songs played is

[tex]\dfrac{C^{10}_2\cdot C^{90}_3}{C^{100}_5}=\dfrac{45\cdot 117,480}{75,287,520}\approx 0.0702[/tex]