Chance

Prologue

1. 0:00Where one ball lands, no one can say. Drop a thousand, and every time, the pile looks much the same.

IThe Galton board

1. 0:10Where will this one land?
2. 0:12At every peg it goes left or right, even odds, like a coin toss.
3. 0:16It went right seven times, so it lands in bin seven. Three more balls take three other paths. Nobody can tell in advance.
4. 0:23Only the number of rights decides the bin, not the order.
5. 0:27Every path is equally likely, so we just count the paths into each bin.
6. 0:31Only one path leads to each end bin. Towards the middle the paths multiply, and more balls land there.
7. 0:38Each ball is still anyone's guess.
8. 0:40But drop enough, and the shares settle: each bin near its share of the paths, a little high here, a little low there.
9. 0:47The middle bin holds just over a fifth, close to what we counted.
10. 0:52From here on, we draw computed chance, not balls.
11. 0:55Double the rows again and again, and shrink the widening board back down: the bars grow thinner, and their tops trace a smooth curve.
12. 1:04Many small, independent lefts and rights, added up, approach this bell curve.

IIRandom walks

1. 1:11Wander like this forever, and where do you end up?
2. 1:14That ball's path, turned on its side, is a random walk.
3. 1:17After a hundred steps, most walks are within ten of the start; after four hundred, within twenty.
4. 1:23Four times the steps, twice the spread. That's a square root.
5. 1:27A different question: how long does each walk spend above zero?
6. 1:31The coin is fair, so intuition says half the time above, half below.
7. 1:36The opposite is true. An even split is the rarest outcome; the most common is lopsided.
8. 1:41Over four in ten walks stay on one side, more than ninety percent of the time. Luck doesn't take turns.
9. 1:48Now a walker on a grid steps at random: one square, in any of four directions.
10. 1:53However many steps it takes, its reach grows only as their square root.
11. 1:57Once the grid blurs away, the wiggles still look much the same.
12. 2:02Where it ends up, no one can say.
13. 2:04Four thousand walkers set off together. No one can predict a single path, yet the crowd spreads into a tidy cloud: densest in the middle, fading smoothly outwards.
14. 2:14On an iron plate, heat spreads like this from a hot spot. Both widen with the square root of time.

IIIBuffon's needle

1. 2:23Why would π turn up in dropped needles?
2. 2:26On a floor ruled with evenly spaced lines, drop a needle as long as the gap. The chance it crosses one is two over π.
3. 2:33Turn it around: drop enough, and twice the number dropped, over the number that cross, homes in on π. This time, five thousand give 3.13.
4. 2:42A needle crosses once or not at all. Its chance of crossing, then, equals its average number of crossings.
5. 2:49Bend it into a noodle, and it can cross one line several times. But it's short needles, end to end: their averages add up, so only the length matters.
6. 2:58Double the length, and the average doubles.
7. 3:03Finally, close each noodle into a circle, as wide as the gap.
8. 3:08Wherever it lands, it crosses exactly twice.
9. 3:13Unrolled, it's π needles long, with two crossings. So one needle averages two over π: its chance of crossing.
10. 3:20π comes from the circle.

IVA random chord

1. 3:24Draw a random chord. Is it longer than a side of the triangle? What are the chances?
2. 3:29Method one: join two random points on the circle. Once the first is fixed, the second must land on the far arc, a third of the circle, for a longer chord.
3. 3:39The answer is one third.
4. 3:41Method two: pick a random radius, then a random point on it. Draw the chord through that point, at right angles to the radius.
5. 3:49If the point is in the inner half, the chord is longer. The answer is one half.
6. 3:53Method three: pick a random point inside the circle, and make it the chord's midpoint.
7. 3:59For a longer chord, it must land in the inner circle: half the radius, a quarter of the area.
8. 4:04Three methods, all called random, and three answers: a third, a half, a quarter.
9. 4:10The trouble is the word random: nobody said what to pick at random.
10. 4:14Try this instead: toss long straws across a table, with no idea where the circle is.
11. 4:19Of the chords the circle cuts, half are longer.
12. 4:22Move the circle, shrink it. The straws don't budge, and it's still about half.
13. 4:27Of the three methods, only method two lets one set of lines, blind to the circle, give every circle the same answer.

VPercolation

1. 4:37If links open with a chance near one half, can the grid connect top to bottom?
2. 4:41Each link gets its own coin toss, all with the same chance of opening. Open links join into clusters.
3. 4:47Even at forty-five percent, it's still all islands.
4. 4:51On an infinite grid, exactly one half is the threshold: at or below it, every cluster is finite; above it, one cluster is infinite.
5. 5:00Just short of one half, suddenly, the largest cluster links top to bottom.
6. 5:05Stop at one half and pull back. The cluster is full of holes within holes, and from farther off, it looks much the same.
7. 5:12Not a solid block, not a thin line, but something in between.
8. 5:15Gold fills over sixty percent of the view; by the end, under thirty.
9. 5:20Pour water in from the top. It can only flow along open links.
10. 5:24At forty-five percent, it soon gets trapped.
11. 5:26Open a few more links, and pour again.
12. 5:29Water through coffee grounds, fire through a forest: each has a threshold like this. Just short of it, nothing gets through; just past it, the way opens.
13. 5:39This time, the water reaches the bottom.

Coda

1. 5:43A bell curve.
2. 5:45A square root.
3. 5:47π.
4. 5:49Half the chords, from scattered straws.
5. 5:51A threshold at exactly one half.
6. 5:53One at a time, no one can say. The order is in the many.