Well, this depends on, in case 2, what does your friend say if the first time, you rolled a 2 and a 5. If he says ‘okay, now you’ve lost the bet’, then it’s the same as case 1. Having rolled the first double, you’ve successfully navigated one of the events of the bet and increased your chances of winning from 1/36 to 1/6, but if you didn’t roll the first double, your chances of winning would decrease from 1/36 to 0. There’s nothing in probability that says your chances can’t change after partially navigating a series of events like this.
If he pulls something else out of his ass after you don’t roll doubles the first time, though, then it’s hard to guess the true probabilities, because you never learn the full rules of the game - your friend is making it up as he goes on.
Just to make this clear… The probability of having 5 boys, if the null hypothesis is true, is 1 in 1*2^5, or 1 in 32. That is 3.125%. Generally in science, 5% is used as the deciding limit (though that’s a somewhat arbitrary convention). Since 3% is less than 5%, we reject the null hypothesis.
IF that is the only observation that has been made! If that couple was the only couple in the world! In fact, we have billions of observations. Now you would have to set this up a bit different. Under the null hypothesis, we would epect 1/32th of couples with 5 children to have 5 boys (and another 1/32th to have 3 girls). Is that what we observe in the population of all those who have 5 children? To give another illustration, you wouldn’t assume who picks 5 stocks correctly is an amazing stock picker without considering the population of all those who attempted to pick stocks and failed.
(I’m taking out the actual statistical calculation part of this… suppose you observe very slightly more that 1/32th, does that mean you can conclude that the chance of having a boy in these circumstances is slightly greater than 50%? Depends on the actual numbers that come out.)
(Also note that depending on your interpretation of the OP, you might instead look at the population of those with 4 children, all of whom are boys, and then measure what comes out for the 5th.)
Suppose you have 100 thousand quarters. Someone tells you that he has a theory, based on where the coins came from, that 40 thousand of them are biased, 20,000 of them will come up heads about 70 percent of the time, and 20 thousand of them will come up tails about 70 percent of the time. If you can verify or disprove this hypothesis, and if it is true identify the biased coins with 99% accuracy (ie, you pick 40,000 coins and if 39,600 or more are really biased)… you get to take all the coins and deposit them in a bank account. You cannot examine the coins in any way but flipping them.
Any notion how many flips of each coin would be required to get accurate data to tackle a problem like this?
This lies outside the realm of simple statistics, like a t-test. This is radically different than determining if all the coins are fair or not. You now have 3 possible hypotheses for any given coin, and the nature of your hypotheses make any averages across the whole population meaningless. Also, you have two completely different tasks – to see if they hypothesis is true, and then apply those learnings to specific coins.
Part I is about proving this hypothesis. Statistics can’t prove hypotheses, only establish if one hypothesis is more likely than another. My gut feeling is that Part I could be achieved by flipping each coin 3 times or so, and then plotting the histogram. If the hypothesis is true, then you should have 3 semi-normal curves superimposed on each other, the pattern would be obvious to see.
(Note how different this is than the OP. In the OP, each coin would have been flipped 4 times already.)
Okay, I’m really interested in this part… can you give me any idea what this histogram would look like?? I tried wiki-ing the term, and got a vague notion of a bar graph with four spaces horizontally, (for 0 heads to 3 heads,) and the columns going up to 50,000 or so. This just doesn’t seem to have enough resolution horizontally to tell us what we need to know, though. (Although if there were more zeroes and threes than we’d expect from a group of completely fair coins, which would be 12,500 of each, that would be an indication that mister Steele’s hypothesis [to assign him an arbitrary name], might be on the right track.)
Yes, you’d probably need more than 4 spaces… this is what I get for thinking on an empty stomach. First, I said semi-normal, because it only approaches normalcy, we acutally have a binomial distribution.
Let’s say you have 4 flips, or 5 spaces. The curve should be centered at 2 heads, should by symmetric, and the proportions should be in line with the binomial distribution. (37.5% at 2 heads, 25% at 1 and 3, 6.25% at 0 and 4).
If the coin is at 70%, you should see it as .8% at 0 heads, 7.5% at 1 head, 26.4% at 2 heads, 41% at 3 heads, 24% at 4 heads. You would see the inverse from the coin at 30% chance of getting heads.
Now, I would think with a graph based on 100,000 coins and 4 flips each, it would be obvious to see whether you basically had a fair population or not. Just looking at the graph and comparing it to how it “should” look will tell you if you have 100,000 fair coins. If it wasn’t fair, would it be precise enough to distinguish that 60% were fair, 20% had a 70% bias, and 20% had a 30% bias? Probably not. I’m not sure how many flips you would need with that. It’s unclear to me what the null and alternative hypotheses are. If the coins aren’t fair, do you assume that Mr. Steele is correct? Or do you need to distinguish Mr. Steele’s claim from the claim that 62,000 are fair, 19,500 have a 72% bias and the remainder have a 24% bias, which would of course require a lot more flips.
Good question about alternative hypotheses, but I don’t have an answer. Maybe I’ll ask Steele when I see him next.
One thing that I was wondering is, especially for this step A testing, if you really need to flip every coin the same number of times, or perhaps you select 1,000 coins at random and flip each of them 20 times. Presumably this has a high chance of giving you some coins from each of the groups, though the proportions from a small sample might not match those of the population as a whole. (On the other hand, if it’s good enough for TV ratings and political opinion polls… )
And yes, given the amount of effort you have to put into it, one would probably be wiser telling mister steele to go stuff all those quarters where the sun don’t shine, compared to even the 25 grand value of them, if you have to keep track of them all and flip each coin many times to accurately guess which group it’s in.
Picture the problem this way: you do a bunch of flips with each coin (how many flips is up for consideration), and plot the average number of heads you get on a histogram. You can imagine that with a very very large number of flips, you’ll see three narrow blips on your histogram - one containing 60% of the coins, at the 0.5 value, and another containing 20% of the coins at 0.3, and another containing 60% of the coins at 0.7. There won’t be any overlap, so you can tell exactly which coins belong in which group.
If you instead plotted the histogram with ten flips each, those histogram lumps would be very wide and would overlap each other to a great extent - you probably couldn’t even tell there were three lumps there.
As you increase the number of flips for each coin, the three lumps on the histogram would get narrower and narrower. Soon you should be able to tell that there are really three lumps, but there would be too much overlap to ID the coins with 99% accuracy. Your question asks how many flips you’d need to do in order for the overlapped area to be less than 1% of the coins.
It’s not the answer, but I thought visualization might help…
You cannot examine the coins in any way but flipping them.
