An event is a selection from the sample space

0:00You already have the list of everything that could happen, and you call it the sample space. Now somebody asks a question in ordinary English, and English is not something you can count. So there is one move that turns the sentence into arithmetic, and it is the whole of this video. The move is this: take the sentence and use it to select from the list. What you select is the event, written E, and formally it is a subset of the sample space.

0:33It can be one outcome or several together, and the whole list and none of it both count. Write it down and the question has stopped being English and started being a set. Toss two coins. The sample space is head-head, head-tail, tail-head, tail-tail. Now the sentence: at least one coin shows a head. Most people write two outcomes here, head-tail and tail-head, and answer one half. Look at what that leaves out. Head-head has two heads on it, and two is at least one.

1:08So the selection is three outcomes, not two, and the answer is three quarters. The misreading costs exactly a quarter, every time. At least one means one or more. Not one and no more. That is the first way the move fails: an element that belongs is dropped. Two more sentences, translated the same way. Roll one die and take the sentence: the number is above four. Above four selects five and six, two outcomes out of six, so the answer is one third.

1:45Now a basket holding an apple, a banana and an orange, and you take one without looking. The sentence: the fruit is yellow. Yellow picks the banana and nothing else, so the event is a single outcome. And that is as far as this one goes, because writing the event and measuring it are two different acts. Hold that thought. It comes back at the end, and it matters. Two selections are worth naming before anything else, because they sit at the two ends.

2:19Select everything. Every outcome satisfies the sentence, so the event is the whole sample space. Something in the list has to happen, so that event has probability one. On an eight-sector spinner numbered one to eight, a number below nine selects all eight sectors, and eight out of eight is one. Now select nothing. No outcome satisfies the sentence, so the event has no elements at all. It is still a perfectly well described event. It simply never happens, and its probability is zero.

2:58Roll one die and ask for a seven: you have described something honestly, and nothing in the list answers to it. Now the number. Once the event is a set its size is a count, and I will write that count n of E. That symbol is my shorthand here, not standard notation, but it makes the rule short. When the outcomes are equally likely, the probability of E is n of E over n of S.

3:29Three of the four two-coin outcomes, so three quarters. Two of the six faces, so one third. Notice what that sentence is carrying. It is not a definition of probability; it is a shortcut with a condition attached. The condition is the phrase equally likely, and if it fails then so does the shortcut. Toss three coins and write out all eight results, from three heads down to none. Now four sentences about the same number, two, and watch four different selections come out.

4:07Exactly two heads selects three of the eight, so three eighths. At least two heads also lets in the three-head result, so four of eight, which is one half. At most two heads throws out only the three-head result and keeps seven, so seven eighths. Not exactly two heads keeps the five that are left, so five eighths. One sample space, one number, four phrasings, and four answers that are all different.

4:37The arithmetic never changed. What changed was which outcomes the sentence let in. Sentences can carry two conditions at once, and it is worth seeing what that does. Take: the first coin is a head, and the heads come to two. Start from the three results with exactly two heads and keep only the ones beginning with a head. Head-head-tail begins with a head. Head-tail-head begins with a head. Tail-head-head does not.

5:09So two of the eight, which is one quarter, down from three eighths. Adding a condition made the answer smaller, and it always will. A second condition can only remove outcomes from a selection, never add them, so the new event sits inside the old one. Sentences that say not are the ones people think need a special rule. They do not. A bag holds three red balls, two blue and one green, and you take one out.

5:41The sentence is: the ball is not red. So select the balls that are not red. Two blue and one green makes three, out of six, which is one half. No rule, no subtraction, just the same move as every other sentence here. Here is another. Five cards carry the letters P, E, A, C and E, and you draw one. Not an E selects P, A and C, three cards out of five.

6:10And P, E or C selects four of the five, because there are two E cards and both of them qualify. There is a shortcut for not, and you may have met it: one minus the probability of the event. One minus three sixths is three sixths, which agrees with the count. So why not just use it? Because it is not free. Ask why one is the right thing to subtract from.

6:39It is right only because the probabilities of all the outcomes in the sample space add up to one, and that is what the shortcut stands on. Watch it fail. Give those six balls a weight of one eighth each instead of one sixth. Now the weights total three quarters, and the two routes disagree by exactly the quarter that is missing. So use the shortcut. Just know you are borrowing something, and know what.

7:10Now back to the yellow banana, because it is where the shortcut runs out. Draw the basket twice. Once holding one apple, one banana and one orange. Once holding one apple, one banana and four oranges. The sentence has not changed. The sample space is still those three names, and the event is still the banana alone. One element out of three, so n of E over n of S says one third, both times.

7:42But in the second basket the banana is one fruit out of six, so its probability is one sixth. Same sentence, same list, same selection, two answers, and counting got one of them wrong by a sixth. Three names in a list does not mean three equal chances. Write the event. Then ask whether you are allowed to divide. Two dice thrown together give thirty-six results, and every one is as likely as every other.

8:15So each single result carries one thirty-sixth. Not one of them carries one. That matters, because a question sometimes asks for an outcome whose probability is one, and here there is no such thing. What can carry a one is an event. The sum lands somewhere between two and twelve selects all thirty-six squares, and that is certain. An event of probability zero is just as easy. The sum is one selects nothing. Both dice show a seven selects nothing.

8:51An outcome can carry a probability of one only when the list has exactly one thing in it. One last thing, and it is the trap that follows every count in this video. Go back to the five cards. Not an E answered three out of five. But write the sample space as the four different letters, P, E, A and C, and not an E becomes three out of four.

9:18Two answers to one sentence. Only the five-card list is equally likely, so only it may be counted in. The same trap is in the bag: six balls gives one half, three colours gives two thirds, wrong by a sixth. So here is the whole of it, in two steps you should keep apart. First, translate: read the sentence and select from the list. The event is an answer in its own right.

9:48Second, and only if the outcomes are equally likely, count: n of E over n of S. Do the first step always. Earn the second one.