tag:blogger.com,1999:blog-7167710173717429664.post7682708883988525895..comments2023-09-04T11:00:41.295-07:00Comments on ToughNut: The Cereal Box SurpriseFreddohttp://www.blogger.com/profile/06503181718327745286noreply@blogger.comBlogger3125tag:blogger.com,1999:blog-7167710173717429664.post-79174758754332230632009-11-22T13:34:12.317-08:002009-11-22T13:34:12.317-08:00Let E(i) = State where there is exactly a collecti...Let E(i) = State where there is exactly a collection of i toys. P(i) = Prob of reaching E(i) from E(i-1).Then P(1) = 1 , P(2) = 4/5. P(3) = 3/5. P(4) = 2/5. P(5) = 1/5. Probability of staying on E(i) is 1-P(i+1).<br />Let A(i) = Expected number of boxes to reach E(i). T(i) = Number of boxes required to reach E(i+1) from E(i)<br /><br />Then . T(i) = P(i+1)(1) + (1-P(i+1))(1+T(i))<br /> => T(i) = 1/{P(i+1)}<br />And thus A(i+1) = A(i) + T(i)<br />=> A(5) = T(0)+T(1)+T(2)+T(3)+T(4)<br />=> A(5) = 1+ 5/4 + 5/3 + 5/2 + 5/1<br />=> A(5) = 137/12<br /><br />Therefore expected expenditure C = 5xA(5) = $57.083randomizerhttps://www.blogger.com/profile/11240734559571229330noreply@blogger.comtag:blogger.com,1999:blog-7167710173717429664.post-73112372722269946452009-11-22T01:50:53.981-08:002009-11-22T01:50:53.981-08:00The above should, of course, be multiplied by 5 (f...The above should, of course, be multiplied by 5 (for $5 per box). Moby Dick.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7167710173717429664.post-34029842903783113212009-11-22T01:42:00.482-08:002009-11-22T01:42:00.482-08:005/5 + 5/4 + 5/3 + 5/2 + 5/1 = 137/12. Moby Dick.5/5 + 5/4 + 5/3 + 5/2 + 5/1 = 137/12. Moby Dick.Anonymousnoreply@blogger.com