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0:00
Hello, I'm Jeffrey Rosenthal.
0:00
I'm a professor of Statistics at the University of Toronto, and this is Stat Support.
0:10
Question from King D. Weeb: Why do statisticians get so worked up over probability?
0:15
Every event is just 50/50; it either happens or it doesn't.
0:18
This is something I've heard before, this idea that, well, if it can either happen or not, it must be 50/50.
0:23
Sometimes that's referred to by philosophers as the principle of indifference, meaning that anything that could happen, they must all have the same probability.
0:30
The thing is, it's just not true.
0:32
When I go home today from the studio, I might get killed by a bolt of lightning or I might not get killed by a bolt of lightning, but I'm pretty sure there's not a 50% chance I'm going to get killed by a bolt of lightning.
0:42
Okay, next we have a question from What the Fuss, who says, "Why is statistics important in life?"
0:51
Really, we're awash in all kinds of different data, so anything from, you know, the spread of disease or crime statistics or, uh, uh, studies of a medical treatment or financial data or public opinion polls, there's so many facts and figures and statistics out there.
0:51
The science of statistics is a way to try to sort through it.
1:06
So if you don't have any statistical knowledge or understanding or perspective, then you're likely to say, "Well, this must be true because my friend said it," or "This must be true because I heard it on the news," or, "I just kind of think it must be true."
1:18
But if you have statistics, you can try to analyze all the facts and figures that are out there and try to see what are the real trends, what's really happening versus what things really aren't the way people think they are.
1:34
Next we have a question from Lawrence ITV, who says, "Question for statisticians: Why did the polls get it so wrong?"
1:36
"Explanations, please."
1:36
Yeah, so public opinion polling, especially when it's predicting elections, is a very high-profile thing, but also a hard thing to do, and usually people notice the mistakes more than the corrections.
1:47
So a lot of public apping for elections has actually been quite accurate and it's predicted things quite well, but there have been some high-profile misses, for example, the US presidential elections of 2016 and 2020.
2:04
Now, even in those cases, typically the poll's prediction compared to the actual results was usually only off by about 4 or 5%, which isn't such a huge amount considering how hard it is to figure out what's going to happen, but it's still a big enough error that if the election's close, it can make a big difference.
2:04
So why is that?
2:15
Well, election polls, of course, they don't ask everybody how they're going to vote, they just ask a sample, usually a few thousand people, and then try to figure out what maybe 100 million people are going to do.
2:25
So that is a challenge.
2:27
The good news is, if the polling is done randomly, that is, we're equally likely to pick every person with the same probability, then we have good statistics to allow us to figure out how accurate we're going to be, what will be the so-called margin of error, you know, how close we'll usually be to the true answer, and actually that works pretty well.
2:46
But what makes it especially hard for the pollsters is that it's hard to get a random sample, and the main reason is because most people don't want to talk to pollsters.
2:57
Polling companies don't necessarily like to talk about it, but their response rates are usually less than 10%, and that can lead to a lot of biases because maybe people who support a certain candidate are a little bit more likely to agree to talk to the pollsters than people who support another candidate, and any little response bias like that can have a huge impact on the results.
3:15
Question from, uh, Come on Matt, Think: "What are some common statistical errors and how can we learn to spot them and, if possible, correct them in others and our own work?"
3:24
One of the biggest things is people don't think about what I like to call the "out of how many" principle, and that's this idea that when something happens at striking, people will compute the probability of it happening in that exact way to that exact person, but not look at the chance that it will happen in some way to somebody.
3:38
There was a woman in England who had two sons who each died in infancy.
3:44
There is something, as you probably know, called SIDS, or Sudden Infant Death Syndrome.
3:52
So maybe just two times she got really, really unlucky and her babies stopped breathing, or maybe she was a murderer and she'd actually, uh, she'd actually suffocated them.
3:52
And she was actually arrested and charged, and at her trial, they said, "Oh, it's so unlikely that there'd be two SIDS cases in the same family that we can rule that out, she must have actually tried to kill them."
4:10
And that's an interesting example where if you just look at the probability given two kids in one family, what's the chance they're both going to die of SIDS?
4:18
Of course, it is very unlikely.
4:19
But then if you say, "Out of all the millions of families in the United Kingdom or in the whole world, what's the chance that somewhere there's a family where two kids both died of SIDS?"
5:04
Extremely likely.
5:04
And it seems like that was the case with her.
5:04
There was actually no other evidence that she had actually tried to kill these kids.
5:04
She was just extremely unlucky, and yet she was convicted, she was jailed, she spent several years in jail before there was enough of an outcry, and eventually on the second appeal, the case was overturned.
5:04
Question from Josh Lev, who says, "What's more likely than winning the lottery?"
5:04
The short answer is everything.
5:04
That is to say, if you're talking about winning a lottery jackpot for one of the big lotteries like Mega Millions or Powerball, then the chance of winning that jackpot with a single ticket is one chance in a couple of hundred million, depending on which lottery, so just incredibly unlikely.
5:04
So compared to that, almost anything you can think of: being killed by a bolt of lightning, or the next person you meet will one day be the President of the United States, or any crazy thing you can come up with, we can estimate the odds for all of them, and they're all more likely than the chance you're going to win the Powerball lottery.
5:17
And in fact, one that I like to use as an example is if you drive to the store to buy your lottery ticket, you're way more likely to be killed in a car crash on your way to the store than you are to win the jackpot.
5:31
Next, we have a question from S. Molly Maul: "I'm just patiently waiting for people to realize that all statistics are skewed because the data is skewed in so many ways that I can't even list them all."
5:31
So not a big fan of statistics, maybe, but that's true.
5:31
There is, that's a good point that all data is going to have some things that are wrong with it.
5:31
Maybe it was biased, maybe it wasn't measured correctly, maybe it only shows part of the story, but I don't think that means we should just forget about it and just forget about statistics and data.
5:31
I think what it means is we have to think carefully when we get data.
5:31
We have to say, "How is this data collected?"
5:31
"Is it an accurate reflection of the truth?"
5:31
"In what ways is it going to be biased or misleading?"
5:31
And then we can still draw inferences from it, but it's true that we have to be careful.
5:31
We have a question from John Fredberg, who says, "About to play what must be the absolute worst casino game in terms of player odds.
5:31
Any guesses?"
5:31
Well, it's an interesting question.
5:31
There's different casinos with different games, but one of the games which, to my surprise, is one of the most popular and also has one of the worst odds against is the, uh, video lottery terminals.
6:30
So people love them, but they usually have at least a 5% and maybe 10% or even 15% house edge, so they're really not the best game.
6:39
Now there are some casino games which have odds which are much better for the player.
6:43
So for example, of the pure chance games, the game craps, where you repeatedly roll a pair of dice, kind of like these, you have a 49.2929% chance of winning.
6:55
Next, a question from Shava Kadzi: "Are murder rates skyrocketing, or the media doesn't have much to report, so they are focusing more on that?"
6:55
Yeah, it's a good question.
6:55
So murder rates have generally been coming down a little bit in the last couple of decades, but in the last few years, there's been a little bit of an uptick, so they're now a little bit higher than they were a few years ago, but they're still quite a bit lower than they were, um, a decade or two ago.
6:55
Also, I've noticed, for example, politicians and police spokespeople and so on, they all will at times say, "Oh, crime rates are way up!" for their own reasons.
6:55
They have reasons for wanting that to be said, even though, you know, maybe it's not actually true.
6:55
So it's just one more reason that if you want to know what's happening with something like, uh, you know, rates of crime, well, don't listen to what a few people are saying, look at the actual statistics, and then you can see the truth.
7:40
Next we have a question from, uh, Brenta Clan, who says, "How does probability work in roulette?"
7:47
So it's a good question.
7:47
Roulettes are fairly simple.
7:47
So the standard American roulette wheel has 38 of those little wedge slots, and two of them are green, there's the zero and the double zero, and then the others are divided into 18 red and 18 black.
7:47
The person at the casino spins the wheel, and presumably it's equally likely to come up any of those 38 different wedges.
8:04
So what it means is if you bet on, for example, red, well, 18 out of the 38 wedges are red, so you have an 18 out of 38 chance of getting red, which is a little bit less than 50%, and that's why if you bet on red, there's an even money payout, but on average, you're going to lose a little bit more money than you win.
8:43
You can also sometimes bet on different things like all the even numbers or something like that, but whichever bet you do, it works out to the same thing, there's a slight edge in favor of the casino, and that's why if you play roulette over a long period of time, it's going to be more and more sure that you're going to lose more money than you win.
8:38
A question from Six Latin Six Lover Six: "Who makes betting odds?
8:43
Is it an algorithm?"
8:43
So it's a really interesting problem for the bookies or the people who are making these, um, odds.
8:43
Now the goal is pretty easy to understand because if you're a bookie, what you want is pretty much to have the same amount of betting on both sides so that in the end, you don't really care if the horse wins or not, or you don't really care if the team wins or not, 'cause either way you're going to make money 'cause you're going to get your cut, whereas if everybody bet on one side and then they all won, then you could lose a lot of money.
9:07
But on the other hand, how they do that is kind of a challenge, and usually they're updating their odds as they go, and if they say, "Oh, everybody's betting on this one team, we better change the odds so that the next bets are more likely to bet on the other side."
9:16
And I'm not a bookie, but my impression is that in the old days, it used to be just kind of by their judgment or, you know, experienced people looking things over and and tweaking things, whereas now there's so much online gambling that a lot of it is automated, and they have algorithms which I think are not simple, based on how everybody's betting and trying to adjust things, but the goal is pretty easy to understand, trying to balance out those bets.
9:16
Question from Zenodot: "What is a stochastic process really?"
9:16
Well, I'm glad you asked.
9:16
So stochastic is just another word for random, so it means random processes or things that proceed randomly in time.
9:16
And the simplest example is actually one I sometimes like to illustrate with my students using a stuffed frog, so I'll do that here.
9:16
And we imagine we have a frog which every second randomly decides either to move one step this way or to move one step this way, and once it does, then the next second it again decides randomly to move one step this way or one step this way, and yet it's actually really interesting for mathematicians to study this.
9:16
What's the chance that the frog will eventually return to where it started?
10:16
Turns out it's 100%, it's certain.
10:18
It might take a really long time, but eventually it's going to return to where it started, and in fact, eventually it's going to be a million steps that way, and eventually it's going to be, um, a billion steps that way.
10:27
It's going to go to every single place eventually if you wait long enough with probability one, we can prove that.
10:39
Next a question from Anael X, who says, "What does it mean to be statistically significant?"
10:39
So statistically significant is saying probably it wasn't just chance, that this is enough of an effect that we can pretty much, you can never do it for sure, but you can pretty much say it's probably not due to chance alone, probably this actually shows something real, there was really a difference or there was really an increase or something really happened, it wasn't just the random luck.
10:57
So the basic idea is pretty simple, it sometimes gets lost in the details, but when you notice something that happens, you know, maybe, oh, this classroom did better on the test than this other classroom, then as statisticians, the fundamental question you're always asking is, does that mean something real, like, oh, maybe the teaching was better in this class or maybe people in that class are, are, you know, smarter, or was it just random luck?
12:55
So you'd never expect any two results to be exactly the same, there's always going to be some differences.
12:55
Okay, next the question from John Elworthy: "Can someone please help with this?
12:55
What are the odds of having three generations of family members being born on the same day?
12:55
First was born on January 10, 1943, the second, same day, 1994, the third, same day in 2022."
12:55
It's actually a good example of the sort of question that there's different ways of looking at the probability.
12:55
So if you just say there's three people, what are the chances they'll all have been born on the same day?
12:55
Well, that's pretty straightforward.
12:55
So you can think, well, the first one could be born on any day, it doesn't really matter.
12:55
Then the second one has roughly one chance in 365 of being born on that same day, and then the third one has roughly one chance in 365 of being born again on that same day.
12:55
So it's one chance in 365 times 365, which, uh, what is that, a little less than one chance in 100,000, I think, so, uh, it's quite unlikely.
12:55
One way I'd like to look at these kind of questions is this is sort of out of how many different ways that this could have happened.
12:55
So even in this one family, probably there's a lot of other people in each of those generations, and if any three of them had matched up their birthdays, then the same tweet could have been written, so right away the chance is a lot bigger because there's lots of different combinations which all could have led to the same conclusion.
12:55
It's not incredible that it happens, but it's still pretty cool when it does happen to you.
12:55
From, says, "How best can a statistician explain P-value to a non-statistician?"
12:55
Yeah, so that's a good question.
12:55
The basic idea of a P-value is the idea of what is the probability that the thing you just observed would have happened just by pure chance if there was no true effect.
12:55
If we look at, let's say, you know, we have some people with a disease and we give them a new treatment, and then a certain number of them get better, do you say, "Oh, well, that means the new treatment really helped"?
12:55
Well, no, 'cause some of them would have gotten better even without this new treatment.
13:03
Maybe more of them got better than you'd expect on average from the new treatment, yeah, but how much more?
13:11
And the P-value question would be, what's the probability if we hadn't given any treatment that that same number or more of the people would still have gotten better?
13:11
And if that P-value is pretty high, you know, maybe there was, there was a 40% chance that they would have gotten better even without the treatment, we haven't really proved anything.
13:11
And the typical standard is that if the P-value is less than 5% or less than one chance in 20, then we say, "Okay, it's pretty unlikely that they all would have gotten better if it hadn't been for this new treatment, so this provides some evidence that the new treatment is helping."
13:11
But if the P-value is larger, it doesn't.
13:11
Okay, so next a question from King Amuso, who says, "Statistically, what are the chances?"
13:07
And right, and this is a, a display of a draw results, and I believe this was from the South Africa Powerball lottery back in December of 2020.
13:51
And what happened was a little surprising, so of the main numbers, there were five numbers chosen in a row, 5, 6, 7, 8, 9, and then the bonus Powerball number chosen was a 10.
14:00
So we had six numbers all in a row for the draw, seemed very surprising.
14:04
So you could say, what are the chances of that happening?
14:06
Well, the rules of the South Africa Powerball then were, you choose five numbers between 1 and 50, and then a bonus number between 1 and 20.
14:14
So you could say, how many different ways could you get them all in a row like that?
14:17
Well, the first five numbers would have to be five numbers in a row starting with something from one, two, three, up to 15, really.
14:26
So that's only 15 ways, and then the Powerball number would have to be the next one, so there's a very small number, and then when you divide that by the total number of different ways you could have chosen those five balls plus the one bonus thing, there's many more of those, so when you divide it, you get that there's a little less than one chance in 2 million that such a sequence like that would have come up.
14:43
Question from, uh, Chris Masterson: "Is it statistically less likely to be in a plane crash if you've already been in one?"
14:50
Well, no, and of course the answer is no.
14:52
If you think about it, how could it be, you know, how could this new plane know, wait a minute, there's somebody on here who was on another crash, so I better not crash this time?
14:59
That's just not the way science works, it's not the way airplanes work, it's not the way pilots work.
15:02
But a lot of people will think that, and the reason people think that is because it's very unlikely any one person is going to be on two different planes that crash, right?
15:11
That's really bad luck, but once you've already been on one, that was very unlucky, but now it doesn't have any effect on the probability of the next plane, they're what we call statistically independent events, so neither one affects the probability of the other.
15:21
So a question from A Tetra says, "Hey, what is the most statistically improbable thing to happen to you?"
15:27
Well, when I was in my early teens, my family went on a trip to Disney World, Florida, and in the middle of it all, we looked up and we saw my father's cousin, Phil, and he lived in Connecticut at the time, and we lived in Toronto, Canada, and we had no idea he was going to be there.
15:46
You say, you know, what are the odds that out of all of the hundreds of millions of people in the United States and all the people that visit Disney World, that my dad's cousin would be there?
15:51
It's a good example that on the one hand, if you just say, what's the chance that one guy would be my dad's cousin Phil, it's incredibly unlikely.
15:58
But as with a lot of things, if you take the bigger picture, you can say, well, my dad's cousin Phil isn't the only person we would have been so surprised to see.
16:04
What about my dad's other cousins, or my mom's cousins, or my cousins, or my piano teacher, or my friend from school, or there's probably a few hundred people that we would have been really surprised to see, and then you say, well, we were at Disneyland for a couple of days, and we went on lots of different rides and so on, and we probably saw thousands of people, and just one of them was my dad's cousin Phil, the other ones were other people, so it's actually not so unlikely.
16:26
And I end up computing there's about one chance in 200 or so, about half of 1%, that if you go on a trip to Disney World and spend a couple of days there on all the rides, that you run into somebody that you know, so it's not so incredible even though it sure was a surprise at the time.
16:39
Okay, so I think that's all the questions for today.
16:41
I hope you learned something, and I hope I'll see you again.