I still don't see how it does. You don't ever have an opportunity to act on that information;
Opportunity to act is irrelevant to how probabilities evolve.
Yesterday you had no numbers in the lottery draw (it hadn't happened yet). The odds that you will win are 1 146 million. Today they have drawn 5 of your balls. There is one ball left. What are the chances you will now win the lottery?
It is a straight forward question. I am not asking, what are the chances of drawing six balls. Some of it has already happened. I am asking what is the odds of drawing one ball from a smaller pool? You have better chances!
Consider another example:
What are the odds of getting two heads in a row? 1 in 4 right?
I flip the coin once and get heads.
What are the chances that this time you will get two heads in a row? That is to say, what are the chances that the next coin flip will produce a head?
The answer is 1 in 2.
Thus. Your probability was 1 in 4 it is now 1 in 2.
I'm going make our lottery into a game similar to poker to show what I am talking about here.
In the lottery I am familiar with the odds of picking a winning ticket is 1 in 13,983,816.
Let's say the jackpot is 10million and it doesn't get shared between players. Every winner gets the maximum jackpot (for ease).
I offer to sell my ticket to you for $10. Is it a good bet?
You pay $10 13,983,815 times and that costs you $139,838,150. You should turn down the offer to buy the ticket. Its not worth it.
Then the draw begins and my ticket has the first of the numbers drawn. At that point I make the same offer to you.
At this stage the odds are different. I only need 5 balls of 5 now. So the odds are 1 in 1,712,304. You pay $17,123,030 over all the trials to win 10million. Not worth it.
If I offer you a ticket with two numbers on it that have been drawn for $10 you should bite my hand off for the offer. The odds of winning the lottery when you already have 2 winning numbers (with only two balls drawn, obviously) is 1 in 178,365 so if you bought the ticket every time you will pay $1,783,640 and you will win $10million. That is a net gain of $8,216,360. That means that your average gain each time is $46. I am offering you $46 for $10. You would be mad to turn down the lottery ticket.
OF course, if you think that the odds are still 1 in 14 million, you will miss a trick here. If you think the odds of winning are still 1 in 14 million when I offer you that ticket partway through the draw, then perhaps we should hook up to play poker sometime.
I guarantee you, if you wrote a computer program that simulated the above situation, you will make a net gain over enough trials. This cannot be the case if the odds of winning are always 14million to 1 against.
It was 1 in 146 million, just like it was for everybody else.
Yes it
was it isn't now.
You're displaying the exact same fallacious thinking that causes people to credit God when they win the lottery - that, somehow, winning is so unlikely that if it happened to you, it had to happen to you
No I'm not. You are misunderstanding me, and I'm sorry for my role in that. If you won, the chances that you won are 1. The chances that you would be the winner are 146 million to 1 against...always.
That's not what probability is about.
Correct. Probability is about working out how likely something is. When you have a winning ticket in your hand, you are very very likely (basically certain) that you have won. When you come across something that is certain we denote the probability of that as being 1.
And, contrary to your assertion, changing the way you learn about the outcome doesn't change how many of those outcomes are desired.
And this is where the misunderstanding is. Hopefully the example above should help you understand that is not what I am saying at all. For each winning number that you get (despite the odds), the odds of you winning increase. That is why, when you have five balls, the chances of you winning
now are about 1 in 50.
It's a hard fact. Try it out. Run the trials. Put yourself in the position of having 5 balls, and pick two random numbers. One number simulates the number you have picked that hasn't been drawn yet, the other number simulates the number drawn. You will find the two numbers coincide about 1 time in 50 (depending on how many balls are left in the lottery). The chances of you getting 5 balls to begin with are very low, but when we know that this outcome has just happened, we don't tally that into our calculations.
I once attended a statistics seminar which had a wonderful example of this:
Everybodies DNA is on record, let's say (not the full string, but the sample that is used in crime detection). And a crime is committed. You are arrested for that crime because your DNA has come up.
The chances of your DNA coming up, says the prosecution, is 1 in a million. Thus - it is likely you have comitted the crime.
You better hope you aren't defending yourself. If I was defending you, I'd counter argue: The chances WERE 1 in a million, but now the event has happened they are 1. However, there are 300 other people in the country who share the same DNA sample as you do. It was a 1 in a million shot that you would have the same DNA sample as the criminal, but now it is a 300 to 1 (against) shot that you are the criminal, since the sample size is 300 and you are just one of those. 300 to 1.
The odds of you being the criminal has changed from 1 in 300 million to 1 in 300.
Other evidence would be needed to be beyond doubt that you were the criminal.
Look, prove me wrong, empirically. Buy a lottery ticket tonight, and then have your friend read you the numbers in whatever way you think will maximize your odds of winning. When you win, you can fly me out to Manchester and I'll buy us a pitcher of Newcastle and tell you I was wrong.
If I buy a lottery ticket the chances of me winning are 1 in 14 million. It doesn't matter how the numbers are read out to me, that doesn't change the odds of me winning the lottery.
IF I bought a lottery ticket and on a 6 in 49 chance I happen to get the first number my chances of winning now are less than 1 in 2 million. If every combination of numbers was out there with no repeats, only 1.75 million of those tickets would have the first number drawn on it.
You can even do the maths backwards. The chances of getting that first number drawn is about 1 in 8. That's another way of saying 1/8 of all combinations has the first number drawn. There are 14 million combinations so 14 million divided by 8 is 1.75 million. Thus my odds of winning are now 1 in 1.75 million. There are only 1.75 million different ways the lottery can now be drawn. One of those ways will see me a winner. That is 1 in 1.75 million.
The chances of me getting the first two numbers are:
6/49 * 5/48 which is about 1 in 80.
IF that does happen then my chances are 14 million divided 80. Or we can say they 1.75 million divided by ten (after getting the first ball, my chances of having the next one is about 1 in 10. 8 * 10 is 80 thus 1 in 80). There are no longer 1.75 million winning tickets. With two numbers drawn there are now 175,000 combinations. My ticket is one of those, which means there is a 1 in 175,000 chance that my ticket will be a winner.
There is now only about a 1 in 12 chance of getting the next ball (I'll leave the maths to you). That means only 1/12 of 175,000 combinations exist. About 15,000 in fact. If I was lucky enough (1 in 12 is fairly lucky), to get the next ball, I have one combination of 15,000 combinations.
A probability of something happening is the number of results you are looking for (1) over the number of possible ways of the event happening (15,000), 1 in 15,000.
The chances of me getting three numbers in the first three balls is about 1 in 960. However, on this occasion we know it has just happened, so we don't work out the odds of that happening and combine it with the odds of getting the next 3 balls. 960 * 15,000 is 14 million (sticking with 2 sig figs). We don't do that. In what are the odds of flipping two heads? 1 in 4. If I flip the coin and get a head, what is the probability of getting a second head? It is not 1 in 4. That is the kind of thinking that gamblers have (It was black three times in a row, it is statistically likely to be red now because four blacks in a row is improbable). Just replace 'black' with 'my number' in the lottery example and we have the same absurd statement. 'I just got three numbers in a row, getting six in a row is a 14 million to 1 shot, thus there is a 14 million to 1 shot of having the next three numbers.'. It's not true at all.
Heads is one outcome out of a sample space of two, so the probability is the same as it was before you flipped it - 1/2. How many different ways do I have to say this before it sticks?
I think considering '1' as a probability might be confusing. Let's try extending the coin flip. Your patient friend flips a coin. He then picks a card from a normal pack of cards. If he picks a Heart and flipped a Tails he will actually say Heads. He does his routine and tells you that it landed on Heads. What are the chances the coin landed on heads? Let's examine it.
Coin Card Friend anounces
H C H
H H H
H S H
H D H
T C T
T H H
T S T
T D T
Since your friend has said Heads, there is a 1 in 5 chance he actually flipped a Tail and picked a Heart (there are five conditions under which he will anounce Heads, one of them is falacious). Thus the chances that he flipped a head is 4 in 5. Not 1 in 2.
If he changes the rules and says he will invert his anouncement when he picks a heart up we get another table:
Coin Card Announce
H C H
H H T
H S H
H D H
T C T
T H H
T S T
T D T
Now he calls Heads. Now there are four ways for him to have said Heads. 3 of them happen when he actually had heads. Thus the chances of him having flipped a heads is 3 in 4. Not 1 in 2.
We can see clearly that the chances of him flipping a heads is 4 in 8 or 1 in 2. But the chances that he has flipped a head given that he says he flipped a head is a different calculation.
Now let's say you flipped a coin and you looked at it and it was Heads. What now? Well there is a small chance that you are wrong. Your eyes are decieving you, maybe you've grown tired of flipping coins, maybe you've lost the plot. Let us say there is a 1 in a billion chance that for whatever reason you think it is a Heads, when in fact it is a Tails. Thus, we can calculate the probability that you flipped a head. Out of a billion, once you get it wrong. Thus the chances you flipped heads are 999,999,999 to 1. Or very nearly 1. Like I said right at the beginning.
Think about it. You flipped a head. You see a head in front of you. There is not a 1 in 2 chance that you flipped a head. There was a 1 in 2 chance it would land on a head. You cannot say 'the coin in front of me that I see as landing on heads could have landed either way, thus it is only a 50% chance of being a head'. Not unless you want to fail basic statistics.
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