1,2

The probability that there are at most k different cards in t drawings is (k/m)^t * m_over_k, where m_over_k means the binomial-coefficient m!/(k!*(m-k)!). This contains also the cases with k-1 different cards, which we want to subtract. Inclusion and exclusion leads to the formula Sum[k=1..m; (-1)^(m-k) (k/m)^t * m_over_k].

a(2)=6 because you have to throw a coin 6 times to get both sides at least once with probability equal or higher than 0.95. (The probability of getting only one side in a series of 6 throws is (1/2)^6 * 2 = 1/32 = 0.03125 < 0.05.)

a(6)=27 because you have to roll a dice 27 times to see all 6 possible outcomes with a probability over 0.95. (If you roll a dice 27 times the probability to get all 6 sides at least once is 0,95658638... . If you roll the dice only 26 times, the probability is 0,94798274... .)

f[1] = 1; f[n_] := f[n] = Block[{k = f[n - 1]}, While[ 2StirlingS2[k, n]*n!/n^k < 19/10, k++ ]; k]; Table[ f[n], {n, 1, 56}]

Cf. A073593 (number of drawings for a 50% probability to see each card, = median) and A060293 (expected value for the number of drawings until each card is drawn once).

Sequence in context: A016861 A145287 A140232 this_sequence A140527 A024730 A024952

Adjacent sequences: A085810 A085811 A085812 this_sequence A085814 A085815 A085816

nonn

Alfred Heiligenbrunner (alfred.heiligenbrunner(AT)aon.at), Jul 25 2003

More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Sep 07 2003