Question

Jane has stored three albums on her iPod. The first album consists of 11 songs by Frank Sinatra, the second album consists of 13 songs by Cliff Richard, including "Congratulations and Celebrations", and the third album consists of 8 songs by John Denver. She programs the shuffle function on her iPod to randomly select 10 songs from the three albums, not repeating any of the songs.

Find the probability that in one run of the shuffle she will not listen to a single song by Cliff Richards.

Find the probability that in one run of the shuffle she will definitely listen to “Congratulations and Celebrations”.

Find the probability that in one run of the shuffle she will definitely listen to “Congratulations and Celebrations”, but no other Cliff Richards song.

Answer 1

Probability = Favourable outcomes by Total Outcomes

Please note ⁿC_r = n! / [(n-r)!*r!]

Total outcomes = choosing 10 songs out of a total of 11 + 13 + 8 = 32 = ³²C₁₀ = 64512240

(a) Favourable outcomes: Excuding all songs of cliff richards. Therefore to choose 10 out of 19 available songs in ¹⁹C₁₀ = 92378

Therefore the required probability = 92378/64512240 = 0.00143

(b) Out of 10 songs 1 song is Congratulations and celebrations. Now the remaining 9 songs can be selected from the remaining 31 songs in ³¹C₉ = 20160075

Therefore the required probability = 20160075/64512240 = 0.3125

(c) Out of 10 songs 1 song is Congratulations and celebrations. Now the remaining 9 songs can be selected from the remaining 11+8 = 19 songs which excludes cliff richards songs in ¹⁹C₉ = 92378

Therefore the required probability = 92378/64512240 = 0.00143