00:01
Here we're defining a series of games, which involves two players, a and b, playing several independent games.
00:11
Independent meaning each game does not depend on the outcome of the previous or any of the previous ones.
00:18
And they keep playing until one player has won two more games than the other player.
00:25
Now we say each of these games, player a wins with probability p and player b wins with probability.
00:31
One minus p.
00:34
So you can see there are no ties because p plus one minus p equals one.
00:41
The first question related to this series of games is what is the probability that exactly four games are played? well, i'm going to claim that there are four possible outcomes, or i'll say four possible ways for this to happen.
01:04
And so i'm going to write each game as a string of letters.
01:08
A if a wins that game and b wins that game.
01:12
So i claim the only possibilities here are a, b, a, b, b, b, b, b, a, b, a, b, a, b, a, b, a, b, a.
01:30
B, a.
01:35
Okay, so you see, if we're playing four games and we end with one player having two more wins than the other one, then we should have one player winning three games and the other player winning one game.
01:48
Now, you may be wondering, i'm missing some cases that look like one player wins three games and then the other one wins one.
01:57
For example, why not a -a -b -a? and the reason we don't include that is because we stop when one player has one more game, two more games than the other player.
02:10
So in fact, if we ever start a string with a, that's it, because player a has won two games and player b is one zero.
02:21
So that will never happen.
02:24
We will never get four games if player a wins the first two games.
02:27
We need to alternate winners in the first two games and then have one player win the subsequent two games.
02:35
Now what's the probability of each of these outcomes? so right here i claim that the probability is p cubed times 1 minus p because the probability of a winning is p and a b winning is 1 minus p.
02:54
In the second case, i claim that the probability is p times 1 minus p cubed.
03:01
And then similarly, we have 3a's and 1b, so the probability is p cubed times 1 minus p.
03:07
And finally the probability here would be p times 1 minus p cubed so the nice one really nice thing about the situation is that actually we can just add up these numbers and we can do that because and maybe i'll do a little aside the probability of let's say x and y should say or here i mean i could it this way but i think it's more instructive to write the probability of probability of x or y is equal to the probability of x plus the probability of x plus the probability of y minus the probability of x and y so here x and y are two events in some probability space maybe they're bad variables because usually you know, x and y are random variables, but i didn't want to confuse it with a and b, usual event names.
04:23
So the probability that one of these outcomes or this, like this outcome or that outcome occurs is going to be the sum of their probabilities minus the probability that they both occur.
04:35
But the nice thing here is the probability that both a, b, a, a, a, b occurs, and a, b, b occurs is zero, right? because we can't have both player a and player b win the third game.
04:49
And so it turns out that we can just add up these probabilities.
04:56
And so we're going to get 2 times p cubed times 1 minus p plus 2 times p times 1 minus p.
05:09
And so if we want to, we can sort of extend this out to write in a nice way.
05:13
So that's 2 p cubed minus 2 p to the 4th plus 2 times.
05:23
And now if we expand out this binomial, we'd get p times 1 minus 3 p times p plus 3 p squared minus p to the 4th.
05:39
And i hope i didn't go too quickly here.
05:42
When i expand this out, the 1 minus p cubed, and i also multiply the p in well doing that.
05:52
And so then we bring it all together, we get 2p cubed minus 2 p to the 4 plus 2p minus 6p squared, plus 6p cubed minus 2 p to the 4.
06:07
And i like to write with the higher powers later, because of course when p is less than 1, then higher power.
06:14
Powers of a number less than one are smaller.
06:17
And so the answer we end up with is 2p minus 6p squared plus 8p cubed minus sorry, it should be 4.
06:37
And so that would be an expression that tells the probability that exactly four games are played.
06:44
Let's just clean up this 4 because i started writing into the 2 before i that was wrong.
06:50
Okay, so that's part a.
06:53
Now let's get on to part b.
06:56
Part b is, well, actually i want to do this, part b would be, what is the probability that a wins the series? okay, so remember from the first part of this question, that for the series to go on a certain length, we need to be alternating winners.
07:25
So essentially, if it's going to go on a long time, we need to have the winner of the first, the winner of the first, game to be different than the winner of the second game.
07:34
Okay, and then going into the third game, we can have any winner, but then the winner of the fourth game needs to be different than the winner of the third game...