10 entries tagged probability, newest first.
A walker that steps at random along a grid comes back to where it started for certain on a line and for certain on a plane — but on a plane 28.25% are still out after 4,000 steps. In three dimensions it comes back only 34.05% of the time, ever. One walk drawn in each, the chance of being home by any step, and why the third dimension is the one where the walker escapes.
A test that is right about 95 in every 100 sick people and 95 in every 100 well ones, used where one person in a hundred is sick, says yes to 590 people in 10,000 — and only 95 of them are sick, so a positive is right 16.1% of the time. A second independent positive takes that to 78.5%. Ten thousand people drawn one square each, with the prevalence, the sensitivity and the specificity on dials.
A fair coin pays 50% on what you stake and takes 40% of it, so staking everything is worth 5% a round on average — and after a hundred rounds the average player holds 131 times the stake while the player in the middle holds one 194th of it. Thirteen players in a hundred end ahead, one in a hundred beats the average, and that one per cent holds 98% of all the money there is. The stake that grows a run fastest is a quarter, break-even is a half, and the second is exactly twice the first for any coin of this kind.
Fill an album of 640 stickers by buying them one at a time and it takes 4,505 stickers on average — but the first half of the album costs 443 of those and the last tenth costs 3,036, because the k-th new sticker takes n over n minus k plus one tries to find. Six faces of a die take 14.7 rolls on average and 13 more often than any other number, and 57 collectors in a hundred finish before the average, at every album size this page draws.
Riffle a deck of 52 cards seven times and it is close to random by the measure Bayer and Diaconis made famous: after seven, the best single bet about the order beats a fair deck by 33 points, and after six by 61. This page works their table out exactly rather than quoting it. Ask instead that every order get at least half its fair chance and it takes eleven, because the deck turned upside down cannot come up at all before the sixth riffle. And eight perfect shuffles put the deck back exactly where it started.
Three ordinary six-sided dice: amber 4 4 4 4 4 9, blue 2 2 2 7 7 7, green 0 5 5 5 5 5. Amber beats blue 7 in 12, blue beats green 7 in 12, and green beats amber 25 in 36 — so there is no best die, and whoever chooses second can always answer the one you chose. Now roll two of each and add. Every one of those three verdicts turns round: blue beats amber 85 in 144, green beats blue 85 in 144, and amber beats green 671 in 1,296. The ring is still a ring and it runs the other way. From three dice each onwards it stops being a ring at all and settles into the order the averages were in the whole time. Two of the three chances are not merely close but the same fraction exactly, at every length; and the two totals can only be equal at all when the number of dice is a multiple of five.
Two people toss a fair coin, a point each time, and you would expect the lead to change hands often and the score to sit near level. Over twenty tosses the likeliest thing that can happen is the opposite: one player leads for every single toss, 35.24 per cent of games, against 6.06 per cent for the even split of ten each. A longer game does not make the lead fairer, it only stops depending on the length. Over ten thousand tosses one player is in front for at least 96 per cent of them in 25.7 games in a hundred, the share stays between 45 and 55 per cent in 6.4, and the lead changes hands about 39 times in the whole game — with no change at all the single likeliest outcome, at any length whatever.
A hundred candidates arrive one at a time, you can only compare them to the ones you have already seen, and every rejection is final. The famous answer is to let 37 go by and then take the first one better than all of them, which lands the very best of the hundred 37 per cent of the time. It also finishes twentieth on average, because in the rounds where no record turns up — 37.4 per cent of them, a different number from the one above and close to it for the same reason — it is left holding whoever walked in last. A different rule, which relaxes what it will accept as the queue runs out, takes the best only 26 per cent of the time and averages rank 3.6 — and never worse than 3.8695 however long the queue.
We each name a run of three coin flips, then flip a fair coin until one of the runs appears. You name yours first — and that is the losing move. Whatever you pick there is an answer that beats it: HHH loses to THH seven times in eight, and the four safest openings still lose two times in three. This page works out all fifty-six ordered pairs in exact fractions by Conway's leading numbers, solves the same fifty-six again by clearing an absorbing Markov chain over rationals, plays two thousand seeded rounds against every opening, and draws the four-sequence cycle that shows why no best sequence exists.
A hundred people each open half the boxes and every one of them must find their own name. Guessing gives one chance in 2^100. Following the loops gives 31.2%, and the number barely moves as the hundred becomes a million: it settles on 1 minus the natural log of two. The argument works because a cycle longer than half the boxes is the only one of its kind — and this page shows exactly where that stops being true.