[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:03.22,Default,,0000,0000,0000,,Let's imagine ourselves in some kind of strange casino
Dialogue: 0,0:00:03.22,0:00:04.80,Default,,0000,0000,0000,,with very strange games
Dialogue: 0,0:00:04.80,0:00:07.46,Default,,0000,0000,0000,,And you walk up to a table, and on that table
Dialogue: 0,0:00:07.46,0:00:09.67,Default,,0000,0000,0000,,there is an empty bag
Dialogue: 0,0:00:09.67,0:00:14.82,Default,,0000,0000,0000,,and the guy who runs the table says, "Look, I've got some marbles here,
Dialogue: 0,0:00:14.82,0:00:19.40,Default,,0000,0000,0000,,three green marbles, two orange marbles, and I'm gonna stick them in the bag
Dialogue: 0,0:00:19.40,0:00:21.47,Default,,0000,0000,0000,,And he literally sticks them into the empty bag
Dialogue: 0,0:00:21.47,0:00:30.44,Default,,0000,0000,0000,,To show you that there is truly three green marbles, and two orange marbles.
Dialogue: 0,0:00:30.44,0:00:35.12,Default,,0000,0000,0000,,And he says, "The game that I want you to play, or if you choose to play,
Dialogue: 0,0:00:35.12,0:00:38.42,Default,,0000,0000,0000,,is you're going to look away, stick your hand in this bag
Dialogue: 0,0:00:38.42,0:00:39.74,Default,,0000,0000,0000,,The bag is not transparent
Dialogue: 0,0:00:39.74,0:00:42.77,Default,,0000,0000,0000,,Feel around the marbles, all the marbles feel exactly the same
Dialogue: 0,0:00:42.77,0:00:45.97,Default,,0000,0000,0000,,And if you're able to pick two green marbles
Dialogue: 0,0:00:45.97,0:00:50.08,Default,,0000,0000,0000,,If you're able to take one marble out of the bag, it's green, you put it down on the table
Dialogue: 0,0:00:50.08,0:00:52.52,Default,,0000,0000,0000,,then put your hand back in the bag
Dialogue: 0,0:00:52.52,0:00:55.40,Default,,0000,0000,0000,,And take another marble, and if that one's also green
Dialogue: 0,0:00:55.40,0:00:58.77,Default,,0000,0000,0000,,Then you're going to win the prize of
Dialogue: 0,0:00:58.77,0:01:06.89,Default,,0000,0000,0000,,You're going to win one dollar if you get two greens.
Dialogue: 0,0:01:06.89,0:01:08.30,Default,,0000,0000,0000,,Well you say, "this sounds like an interesting game,
Dialogue: 0,0:01:08.30,0:01:10.39,Default,,0000,0000,0000,,How much does it cost to play?"
Dialogue: 0,0:01:10.39,0:01:15.27,Default,,0000,0000,0000,,And the guy tells you it is 35 cents to play.
Dialogue: 0,0:01:15.27,0:01:18.39,Default,,0000,0000,0000,,So obviously, fairly low stakes casino.
Dialogue: 0,0:01:18.39,0:01:23.07,Default,,0000,0000,0000,,So my question to you is, would you want to play this game?
Dialogue: 0,0:01:23.07,0:01:25.29,Default,,0000,0000,0000,,And don't put, you know, the fun factor into it
Dialogue: 0,0:01:25.29,0:01:32.35,Default,,0000,0000,0000,,Just economically, does it makes sense for you to actually play this game?
Dialogue: 0,0:01:32.35,0:01:35.37,Default,,0000,0000,0000,,Well let's think through the probabilities a little bit.
Dialogue: 0,0:01:35.37,0:01:41.22,Default,,0000,0000,0000,,So first of all, what's the probability that the first marble you pick is green?
Dialogue: 0,0:01:41.22,0:01:47.94,Default,,0000,0000,0000,,What's the probability that first marble is green?
Dialogue: 0,0:01:47.94,0:01:50.14,Default,,0000,0000,0000,,Actually, just let me write first green
Dialogue: 0,0:01:50.14,0:01:54.41,Default,,0000,0000,0000,,Probability first green
Dialogue: 0,0:01:54.41,0:01:57.15,Default,,0000,0000,0000,,Well, the total possible outcomes
Dialogue: 0,0:01:57.15,0:01:59.19,Default,,0000,0000,0000,,There's 5 marbles here all equally likely
Dialogue: 0,0:01:59.19,0:02:01.30,Default,,0000,0000,0000,,So there's 5 possible outcomes
Dialogue: 0,0:02:01.30,0:02:04.52,Default,,0000,0000,0000,,3 of them satisfy your event that the first is green
Dialogue: 0,0:02:04.52,0:02:08.60,Default,,0000,0000,0000,,So there's a three-fifths probability that the first is green.
Dialogue: 0,0:02:08.60,0:02:10.44,Default,,0000,0000,0000,,So you have a three-fifths chance
Dialogue: 0,0:02:10.44,0:02:12.46,Default,,0000,0000,0000,,Three-fifths probability, I should say
Dialogue: 0,0:02:12.46,0:02:15.89,Default,,0000,0000,0000,,That after that first pick you're kind of still in the game.
Dialogue: 0,0:02:15.89,0:02:20.81,Default,,0000,0000,0000,,Now, what we really care about is your probability of winning the game.
Dialogue: 0,0:02:20.81,0:02:24.70,Default,,0000,0000,0000,,You want the first to be green, and the second green.
Dialogue: 0,0:02:24.70,0:02:27.61,Default,,0000,0000,0000,,Well let's think about this a little bit. What is the probability
Dialogue: 0,0:02:27.61,0:02:32.11,Default,,0000,0000,0000,,that the first is green
Dialogue: 0,0:02:32.11,0:02:32.96,Default,,0000,0000,0000,,I'll just write "g" for green
Dialogue: 0,0:02:32.96,0:02:38.20,Default,,0000,0000,0000,,And the second is green.
Dialogue: 0,0:02:38.20,0:02:41.26,Default,,0000,0000,0000,,Now, you might be tempted to say
Dialogue: 0,0:02:41.26,0:02:43.56,Default,,0000,0000,0000,,"Oh well the second being green is the same probability,
Dialogue: 0,0:02:43.56,0:02:47.61,Default,,0000,0000,0000,,it's three-fifths. I can just multiply three-fifths times three-fifths
Dialogue: 0,0:02:47.61,0:02:49.36,Default,,0000,0000,0000,,And I'll get nine over twenty-five
Dialogue: 0,0:02:49.36,0:02:52.24,Default,,0000,0000,0000,,Seems like a pretty straight-forward thing."
Dialogue: 0,0:02:52.24,0:02:56.18,Default,,0000,0000,0000,,But the realization here is what you do with that first green marble.
Dialogue: 0,0:02:56.18,0:03:00.04,Default,,0000,0000,0000,,You don't take that first green marble out, look at it, and put it back in the bag.
Dialogue: 0,0:03:00.04,0:03:05.56,Default,,0000,0000,0000,,So when you take that second pick, the number of green marbles that are in the bag
Dialogue: 0,0:03:05.56,0:03:07.36,Default,,0000,0000,0000,,depends on what you got on the first pick.
Dialogue: 0,0:03:07.36,0:03:09.04,Default,,0000,0000,0000,,Remember, we take the marble out
Dialogue: 0,0:03:09.04,0:03:11.43,Default,,0000,0000,0000,,if it's a green marble or whatever marble it is
Dialogue: 0,0:03:11.43,0:03:14.18,Default,,0000,0000,0000,,Whatever after the first pick, we leave it on the table.
Dialogue: 0,0:03:14.18,0:03:17.41,Default,,0000,0000,0000,,We are not replacing it, so there's not any replacement here.
Dialogue: 0,0:03:17.41,0:03:20.18,Default,,0000,0000,0000,,So these are not independent events.
Dialogue: 0,0:03:20.18,0:03:24.56,Default,,0000,0000,0000,,Let me make this clear, not independent.
Dialogue: 0,0:03:24.56,0:03:30.02,Default,,0000,0000,0000,,Or in particular, the second pick is dependent on the first.
Dialogue: 0,0:03:30.02,0:03:36.89,Default,,0000,0000,0000,,Dependent on the first pick.
Dialogue: 0,0:03:36.89,0:03:41.68,Default,,0000,0000,0000,,If the first pick is green, then you don't have three green marbles in a bag of five
Dialogue: 0,0:03:41.68,0:03:47.94,Default,,0000,0000,0000,,If the first pick is green, you now have two green marbles in a bag of four
Dialogue: 0,0:03:47.94,0:03:51.60,Default,,0000,0000,0000,,So the way that we would refer to this is the probability of both of these happening
Dialogue: 0,0:03:51.60,0:03:58.72,Default,,0000,0000,0000,,Yes, it's definitely equal to the probability of the first green
Dialogue: 0,0:03:58.72,0:04:07.18,Default,,0000,0000,0000,,times, now this is kind of the new idea, the probability of the second green
Dialogue: 0,0:04:07.18,0:04:10.06,Default,,0000,0000,0000,,given, this little line over here
Dialogue: 0,0:04:10.06,0:04:12.89,Default,,0000,0000,0000,,just this straight up, vertical line just means given
Dialogue: 0,0:04:12.89,0:04:16.42,Default,,0000,0000,0000,,Given, this means given
Dialogue: 0,0:04:16.42,0:04:19.11,Default,,0000,0000,0000,,Given that the first was green.
Dialogue: 0,0:04:19.11,0:04:25.71,Default,,0000,0000,0000,,Now what is the probability that the second marble is green given that the first marble was green?
Dialogue: 0,0:04:25.71,0:04:28.18,Default,,0000,0000,0000,,Well we drew this scenario right over here
Dialogue: 0,0:04:28.18,0:04:33.26,Default,,0000,0000,0000,,If the first marble is green there are four possible outcomes
Dialogue: 0,0:04:33.26,0:04:34.64,Default,,0000,0000,0000,,not five anymore
Dialogue: 0,0:04:34.64,0:04:39.18,Default,,0000,0000,0000,,And two of them satisfy your criteria.
Dialogue: 0,0:04:39.18,0:04:41.51,Default,,0000,0000,0000,,So two of them satisfy your criteria.
Dialogue: 0,0:04:41.51,0:04:46.26,Default,,0000,0000,0000,,So the probability of the first marble being green and the second marble being green
Dialogue: 0,0:04:46.26,0:04:48.40,Default,,0000,0000,0000,,Is going to be the probability that your first is green
Dialogue: 0,0:04:48.40,0:04:50.30,Default,,0000,0000,0000,,So it's going to be three-fifths
Dialogue: 0,0:04:50.30,0:04:54.28,Default,,0000,0000,0000,,Times the probability that the second is green given the first was green.
Dialogue: 0,0:04:54.28,0:04:58.47,Default,,0000,0000,0000,,Now you have one less marble in the bag and we're assuming that the first pick was green
Dialogue: 0,0:04:58.47,0:05:01.89,Default,,0000,0000,0000,,So you only have two green marbles left.
Dialogue: 0,0:05:01.89,0:05:05.17,Default,,0000,0000,0000,,And so what does this give us for our total probability?
Dialogue: 0,0:05:05.17,0:05:07.14,Default,,0000,0000,0000,,Let's see. Three-fifths times two-fourths
Dialogue: 0,0:05:07.14,0:05:09.34,Default,,0000,0000,0000,,well two-fourths is the same thing as one half
Dialogue: 0,0:05:09.34,0:05:14.20,Default,,0000,0000,0000,,This is going to be equal to three-fifths times one half
Dialogue: 0,0:05:14.20,0:05:16.05,Default,,0000,0000,0000,,Which is equal to three tenths
Dialogue: 0,0:05:16.05,0:05:20.94,Default,,0000,0000,0000,,Or we could write that as zero point three zero
Dialogue: 0,0:05:20.94,0:05:25.29,Default,,0000,0000,0000,,Or we could say that there is a 30 percent chance
Dialogue: 0,0:05:25.29,0:05:29.44,Default,,0000,0000,0000,,of picking two green marbles when we are not replacing.
Dialogue: 0,0:05:29.44,0:05:31.97,Default,,0000,0000,0000,,So, given that, let me ask you the question again
Dialogue: 0,0:05:31.97,0:05:35.05,Default,,0000,0000,0000,,Would you want to play this game?
Dialogue: 0,0:05:35.05,0:05:38.63,Default,,0000,0000,0000,,Well if you played this game many, many, many, many times
Dialogue: 0,0:05:38.63,0:05:43.87,Default,,0000,0000,0000,,On average, you have a 30 percent chance
Dialogue: 0,0:05:43.87,0:05:47.47,Default,,0000,0000,0000,,of winning one dollar.
Dialogue: 0,0:05:47.47,0:05:49.02,Default,,0000,0000,0000,,And we haven't covered this yet,
Dialogue: 0,0:05:49.02,0:05:52.05,Default,,0000,0000,0000,,So your expected value is really going to be
Dialogue: 0,0:05:52.05,0:05:55.69,Default,,0000,0000,0000,,30 percent times one dollar
Dialogue: 0,0:05:55.69,0:05:56.75,Default,,0000,0000,0000,,This gives you a little bit of a preview
Dialogue: 0,0:05:56.75,0:06:01.46,Default,,0000,0000,0000,,Which is going to be thirty cents
Dialogue: 0,0:06:01.46,0:06:03.02,Default,,0000,0000,0000,,Thirty percent chance of winning one dollar
Dialogue: 0,0:06:03.02,0:06:04.66,Default,,0000,0000,0000,,You would expect, on average,
Dialogue: 0,0:06:04.66,0:06:06.15,Default,,0000,0000,0000,,if you were to play this many, many, many times
Dialogue: 0,0:06:06.15,0:06:11.00,Default,,0000,0000,0000,,that playing the game is going to give you 30 cents.
Dialogue: 0,0:06:11.00,0:06:13.05,Default,,0000,0000,0000,,Now, would you want to give someone
Dialogue: 0,0:06:13.05,0:06:17.66,Default,,0000,0000,0000,,35 cents to get on average 30 cents?
Dialogue: 0,0:06:17.66,0:06:21.02,Default,,0000,0000,0000,,No! You would not want to play this game.
Dialogue: 0,0:06:21.02,0:06:23.64,Default,,0000,0000,0000,,Now, one thing I will let you think about is
Dialogue: 0,0:06:23.64,0:06:24.92,Default,,0000,0000,0000,,Would you want to play this game
Dialogue: 0,0:06:24.92,0:06:29.29,Default,,0000,0000,0000,,If you could replace the green marble the first pick
Dialogue: 0,0:06:29.29,0:06:31.49,Default,,0000,0000,0000,,After the first pick if you could replace the green marble
Dialogue: 0,0:06:31.49,0:06:36.82,Default,,0000,0000,0000,,Would you want to play the game in that scenario?