00:01
All right, so for this problem, the first part says that there's an earn with four balls, three that are losers, and one that's a winner.
00:11
And we're taking turns drawing, and we want the probability that we who draw first wins.
00:18
So the probability that we win is equal to the probability that the first ball drawn is the winner, plus the probability that the third ball drawn is a winner, because we're taking turns.
00:31
And so if we don't win on the first try and our partner doesn't win on their first try, then we can also win on the third draw.
00:39
And the probability that we lose is equal to the probability that the winner is selected on the second draw, plus the probability that the winner is selected on the fourth draw.
00:49
So the probability that the winning ball is selected on the first draw is just equal to 1 out of 4 because there's one winner out of 4 total.
00:57
And the probability that the ball is drawn, the winning balls drawn on the second is equal to that probability that it is not drawn on the first, which is three out of four, times the probability that it's drawn on the second, which is one out of three.
01:10
There's one winning ball out of three remaining balls.
01:13
And we can do the same thing for the probability that the winning ball is run on the third.
01:19
It's equal to three -fourths times the probability that it wasn't drawn on the second, which is two -thirds, times now there's two balls remaining, one of which is the winner.
01:28
And finally, we can do the same thing here.
01:32
And now there's a one -half chance that the loser is drawn on the third draw, but then it would be guaranteed that it's drawn on the fourth draw.
01:39
And so you can see that all these actually come out to be one -fourth.
01:53
And so the probability that we win is just equal to one -fourth plus one -fourth, which is equal to one -half, if the drawing is done without replacement.
02:02
However, if we add one ball, so now there's five balls with one winner and four losers, and we do the drawing with replacement, what is the probability that we win if we draw in the first try? well, we know that let's let k be the number of trials required to win.
02:38
So if k equals 1, then the first draws the winner.
02:42
If k equals 3, then the third draws the winner, etc.
02:46
So, or actually, sorry, trials per person.
02:54
If i went on my first draw, that means that k equals one.
02:59
And if my partner wins on their first trial, that means k is also equal to one for him.
03:06
And let's let's let's let's then be the total number of trials, including the win.
03:17
So the probability that x is equal to a given number k is going to be equal to 0 .8, or the probability that we draw a loser times k, minus 1 times 0 .2 to the first power, meaning that we're losing k minus 1 times, and we're winning that last time 1.
03:50
And so now if we take k as a total number of trials per person, say that we want to win on our 10th trial.
04:00
This is just arbitrary.
04:02
Then that means that we need to lose 10 times, or we need to lose nine times...