Complements: knowing one probability hands you the other

0:00Before any rule, a pattern. A coin. The chance of a head is a half, and the chance of a tail is a half. Those two add to one. A die. The chance of a result above four is a third, and the chance of four or below is two thirds. Those two add to one as well. Two completely different experiments, four separate answers, each one reached by its own count - and both pairs land on one.

0:30That is worth being suspicious about, because nobody arranged it. So look at what the pairs have in common. In each one, the second event is the first event denied. Not tails as a separate thing you happened to also ask about. Tails is what is left when heads does not happen. Four or below is not a new question. It is what is left when the result is not above four.

0:59That relationship has a name. The second event is called the complement of the first. And the two together are called complementary events. Notice which way round that goes. The complement is not the leftover unlikely bit - it is simply everything else, and it can be very much the larger side. Since the idea comes up constantly, it gets a symbol. Call the event E. Its denial is written as E with a bar drawn over it, and read as not E.

1:35The bar is a name, not an instruction. It does not say subtract anything. It says: the event made of every outcome E leaves out. The arithmetic comes later, and it comes from the outcomes, not from the notation. Now the reason the pattern happened, and it needs no new principle at all. Draw the outcome list as a single bar, one piece per outcome. The event E takes some of those pieces. Not E takes exactly the ones E left.

2:09So every outcome is on one side of the cut or the other. None is missing, and none is on both sides. Which means the two counts add up to the whole count - the numerators fill the denominator between them. And that is the fact from the topic on totals: a group of events that covers everything without overlapping adds up to one. A pair of complementary events is the smallest possible example of that group. Two pieces instead of many.

2:43So the probability of E plus the probability of not E is one, and the reason is the picture rather than the arithmetic. Now rearrange it, because the rearranged form is the one that earns its keep. If the two add to one, then each is one minus the other. The probability of not E is one minus the probability of E. Written that way it stops being an observation and becomes a tool: it converts one answer into the other with a single subtraction.

3:19And it works in whichever direction you need. Know either one, and you have both. One warning while it is on the board. Once is always enough. Subtracting a second time takes you straight back to where you started, which is a mistake worth naming before it happens. Time to use it, and the first case is one where you can watch the two routes agree. A shuffled pack of fifty-two cards. Draw one.

3:49The chance it is an ace: four aces out of fifty-two, which is one thirteenth. Now the chance it is not an ace, counted directly. Fifty-two cards minus the four aces leaves forty-eight. Forty-eight over fifty-two, which is twelve thirteenths. And by subtraction: one minus one thirteenth is twelve thirteenths. Same answer, and it had to be, because the forty-eight cards were the fifty-two with the four taken out. Notice that the complement here is the huge side and the event is the small one. Nothing about a complement makes it small.

4:31Notice too that neither route was much work. The rule is not always a shortcut, and here it is a check. Now a case where the rule is not a convenience but the only way through. Two players, Nina and Petra, play a tennis match. The chance that Nina wins is nought point six two. What is the chance that Petra wins? There is no list of outcomes to count here. Nobody has told you how the match works.

5:03But if Petra winning is exactly Nina not winning, then the two chances add to one. So Petra wins with probability one minus nought point six two, which is nought point three eight. One number in, one number out, and no counting anywhere. That step was quick, and it quietly assumed something. Say it out loud. It assumed there are only two ways the match can end. That is true of tennis, because a tennis match is played until somebody wins. There is no third result.

5:39But the moment a third result exists, the subtraction is simply wrong. Take a game that can be drawn. Now the bar has three pieces, not two: the first player wins, the second player wins, or nobody does. If the first player wins with nought point six two and a draw happens one time in twenty, the second player wins with nought point three three - not nought point three eight.

6:08One minus nought point six two overstates her by exactly the chance of the draw, and it would do that for any draw you like. So the rule always needs its licence: the two events must cover everything and share nothing. Sounding like opposites is not enough. Here is a case where the complement is chosen deliberately, to avoid work. Two friends, Clara and Emma. Leap years set aside, treat the three hundred and sixty-five days as equally likely for the second birthday.

6:45What is the chance the two birthdays are different? Whatever day the first birthday falls on, three hundred and sixty-four of the days leave the second one different. So three hundred and sixty-four over three hundred and sixty-five. Now the same birthday. That is the denial, so it is one minus three hundred and sixty-four over three hundred and sixty-five, which is one over three hundred and sixty-five. And notice which side got counted directly.

7:19The awkward one to argue about was the small one, so it was left to the subtraction and the easy side was counted. That choice - count whichever side is easier, subtract for the other - is most of the practical value of this rule. Which brings us to the phrase that should make you reach for the complement every single time: at least one. Two coins that can be told apart, tossed together. Four outcomes: head head, head tail, tail head, tail tail.

7:54The chance of at least one head. Counted directly: head head qualifies, head tail qualifies, tail head qualifies. Three of the four. Three quarters. Counted the other way: the denial of at least one head is no head at all, which is one outcome, tail tail. That is a quarter, and one minus a quarter is three quarters. Three things to check, or one thing to check. On four outcomes it hardly matters. Watch what happens when the experiment grows.

8:29With eight coins there are two hundred and fifty-five favourable outcomes and still exactly one unfavourable one. Across coin experiments from one coin up to eight, counting the favourable side directly means listing five hundred and two outcomes. Counting the denials means listing eight. Now the mistake this phrase invites. The denial of at least one head is not at least one tail. At least one tail is also three quarters, and both of them happen on head tail. Two events that can happen together are not each other's denial.

9:09The denial of at least one is always none at all. One more: a die thrown twice. Thirty-six outcomes, and the chance of a five appearing at least once. Strike out the row and the column that contain a five, and a five by five block of twenty-five is left standing - those are the throws with no five anywhere. Twenty-five over thirty-six for no five, so eleven over thirty-six for at least one.

9:41Two last forms, because questions come at this from both sides. First: you are told a probability is nought point nought five, and asked for the probability of the event not happening. One minus nought point nought five is nought point nine five. Second: you are told that the chance two of three students do not share a birthday is nought point nine nine two, and asked for the chance that two of them do share one.

10:12One minus nought point nine nine two is nought point nought eight. Those two questions feel like mirror images, and the arithmetic is identical: take what you were given away from one. What differs is only which member of the pair the question chose to hand over. The trap is to feel that the second one is reversed and needs undoing - and to subtract a second time, which turns nought point nought eight straight back into nought point nine nine two.

10:45Across every probability you can build on an experiment with up to eight outcomes, subtracting twice and subtracting once give different answers on four hundred and twelve of the five hundred and ten. They agree only on the ninety-eight where the chance is exactly a half, and where it does not matter. So, the whole thing. The denial of an event is built from precisely the outcomes the event leaves behind, so the two fill the list with no gap and no overlap, and their probabilities add to one.

11:24Rearranged, that is one subtraction, and it is worth reaching for whenever the far side is the easier one to count - which is nearly always what at least one means. But the licence is the whole thing. The two events must cover everything, and they must share nothing. It is tempting to treat adding to one as proof of that, and it is not. Take every pair of events you can build on an experiment with up to five outcomes - one thousand three hundred and sixty-four pairs.

11:58Sixty-two of them are genuinely complementary. Three hundred and fifty of them have probabilities adding to one. So two hundred and eighty-eight pairs add to one without being complementary at all. Fewer than one pair in five that totals one has earned the name. Adding to one is what the condition buys you. It is not evidence that you have it. Check the outcomes, not the arithmetic. Then subtract.