Hillary Clinton is Done

[Skeptic Ginger]

This is exactly why you can use the geometric distribution and get a 1/64 odds - each is a so-called independent Bernoulli trial. You can also use the Binomial distribution to find the odds of any particular amount of each outcome for a set of 6 coin tosses; you will find that 3 is the expected value.

Here is why: consider using 6 different coins for clarity's sake. You like them up in a set order and write out the outcome, e.g. HTHHTT.

Now, by combinatorics, there are 2^6 = 64 possible combinations. But not all of these represent different outcomes in terms of number of heads. For example, HHHHHT and HTHHHH both give the outcome "5 heads, 1 tails."

The probability of any outcome then, is the number of unique combinations representing that outcome, over the total number of unique combinations (the binomial distribution calculates exactly this.) For example, for the above 5 Heads, 1 Tails outcome, we may represent it with six different combinations (the single tails can occur in any one of six places), and so the probability of that is 6/64 or 3/32 ~ 1/11.

For all heads, or all tails however, we are in the unique position of having only one possible combination representing each: HHHHHH and TTTTTT. Thus, either of those has only a 1/64 chance of occurring. By comparison, there are 32 different ways to get 3 heads, 3 tails.

I hope this is clear enough. This is really just an elaboration on the principle of multiplication, a basic statistical law easily derived through the definition of independent events.

I'm sure we are just talking about the elephant's trunk and the elephant's tail, because I know you guys know your stuff, but chroot (Warren) from the physicsforums is where I grew my understanding.

The coin flips are entirely independent. Every time you flip a coin, you have a 50-50 chance of getting both heads or tails, no matter what happened before....

Now let's look at the possibilities with 10 spins: All blacks (1 possibility) Some mixture of blacks and reds (1,022 possibilities) All reds (1 possibility) Do you see what's happening here? There are 1,024 possibilites, only two of which are all black or all red. That leads people -- including yourself -- into thinking that a sequence of all reds or all blacks is highly improbable. It's not! The sequence BBBBBBBBBB is no less probable than the sequence RBRRBRBRBB. Sure, one looks more "random," but, in fact, both sequences have exactly the same probability. Every possible sequence of spins has a 1/1,024 chance of happening....

Sometimes people refer to "the law of large numbers" when dealing with probabilities. Only if you flip the coin a large number of times can you be certain of getting 50% heads and 50% tails. If you flip it just once, obviously you don't -- you get either 100% heads or 100% tails. Only if you flip the coin an infinite number of times, in fact, are you guaranteed of getting 50% heads and 50% tails.

 

[TubbaBlubba]

I'm sure we are just talking about the elephant's trunk and the elephant's tail, because I know you guys know your stuff, but chroot (Warren) from the physicsforums is where I grew my understanding.

Here is the thing: we don't care about the order. Each sequence is equally probable, but some sequences represent the same outcome. And not all outcomes have the same amount of possible sequences.

 

[Foolmewunz]

So, what are the odds that Gowdy's committee would leak six sets of email in a row? (Apparently, like these six coin tosses, it's now 1:1.)

 

[Childlike Empress]

Here is the thing: we don't care about the order. Each sequence is equally probable, but some sequences represent the same outcome. And not all outcomes have the same amount of possible sequences.

In other words, the guy is misleading you, Skeptic Ginger, by comparing the outcome RBRRBRBRBB to BBBBBBBBBB. Those odds are the same, but in our case, for Bernie to win three of the six coin tosses, he could win the first three and lose the last three, which gives BBBHHH - same odds as Hillary winning all. But if he loses the first three and wins the last three, HHHBBB, it's still 3:3. Two events of 64 possible. Now they win in alternating order - Bernie starts, BHBHBH, or Hillary starts, HBHBHB. Another two events where the outcome is 3:3 victories. Makes it 4. I'll leave it to you to find the other 28.

 

[TubbaBlubba]

So, what are the odds that Gowdy's committee would leak six sets of email in a row? (Apparently, like these six coin tosses, it's now 1:1.)

And on six coin tosses, it's 1 in 64. Yeah, the odds speak for themselves, but it's a high enough probability to be a perfectly mundane outcome, compared to the odds of rigging the tosses.

Now, if there had been 25 or 100 coin tosses, then shenanigans would have been far more likely than a perfect run.

 

[sunmaster14]

Don't get what you mean. The coin flip is always for one, when it occurs. At least according to my understanding.

Each precinct delegate is worth approximately 1/8th of a state-wide convention delegate, and (statistically) an even smaller fraction of a national convention delegate. So each coin toss, which was at most deciding one precinct delegate, was only deciding approximately 1/8th of a "real" delegate, and far less of a national convention delegate. As far as I can tell (and contrary to the NPR article which was cited), there were only six coin tosses to decide an extra precinct delegate, and Clinton won all six. A lucky outcome for her, but not particularly unlikely. Remember, if there hadn't been an anomalous result, we wouldn't have heard about it. Availability bias at work!

In the end, it mattered very little because Clinton's luck garnered an extra 3 precinct delegates over her expected number, and 3 out of 11,065 rounds to zero in any delegate count that actually matters.

 

[sunmaster14]

This is exactly why you can use the geometric distribution and get a 1/64 odds - each is a so-called independent Bernoulli trial. You can also use the Binomial distribution to find the odds of any particular amount of each outcome for a set of 6 coin tosses; you will find that 3 is the expected value.

Here is why: consider using 6 different coins for clarity's sake. You like them up in a set order and write out the outcome, e.g. HTHHTT.

Now, by combinatorics, there are 2^6 = 64 possible combinations. But not all of these represent different outcomes in terms of number of heads. For example, HHHHHT and HTHHHH both give the outcome "5 heads, 1 tails."

The probability of any outcome then, is the number of unique combinations representing that outcome, over the total number of unique combinations (the binomial distribution calculates exactly this.) For example, for the above 5 Heads, 1 Tails outcome, we may represent it with six different combinations (the single tails can occur in any one of six places), and so the probability of that is 6/64 or 3/32 ~ 1/11.

For all heads, or all tails however, we are in the unique position of having only one possible combination representing each: HHHHHH and TTTTTT. Thus, either of those has only a 1/64 chance of occurring. By comparison, there are 32 20 different ways to get 3 heads, 3 tails.

FTFY. Otherwise, a very nice explanation.

The probability of getting exactly K heads (or tails) out of N flips is {N choose K}, which is N*(N-1)*(N-2)*...*(N-K+1)/K!. Note that K! = K*(K-1)*...*2*1.

So, for 6 flips, the # of ways of getting exactly 0, 1, 2, 3, 4, 5, or 6 heads is 1, 6, 15, 20, 15, 6, or 1, respectively. The symmetry of course is obvious because getting a head is equivalent to failing to get a tail, and failing to get a head is equivalent to getting a tail. This actually proves that {N choose K} is equal to {N choose (N-K)}. Also, the sum of {N choose K} for all K between 0 and N, inclusive, is equal to 2^N.

 

[wareyin]

The math is interesting, here, but the reports I'm seeing say Sanders won 5 of the 6 coin flips, not that Clinton won them all. http://www.cnn.com/2016/02/02/politics/hillary-clinton-coin-flip-iowa-bernie-sanders/index.html

 

[elbe]

The math is interesting, here, but the reports I'm seeing say Sanders won 5 of the 6 coin flips, not that Clinton won them all. http://www.cnn.com/2016/02/02/politics/hillary-clinton-coin-flip-iowa-bernie-sanders/index.html

The article says Sanders won 6 of 7 coin flips (5-1 against Clinton, 1-0 against O'Malley) that CNN could verify through some MS app but admitted that nearly half of the precincts weren't using the app that they could check records on and thus didn't know how many coin flips occurred within them or what the results of them might have been. And then it goes into how the coin flips are for county delegates, not state delegates, and how the flips should have had minimal impact on the state results.

 

[Childlike Empress]

FTFY.

Ah, and I copied it while thinking "is it really that high?". Laziness.

btw, now checked the context of what SG quoted - 11 years ago they were discussing the question if the odds of a single (last) toss would be influenced by what has already happened, which of course they aren't.

Explained this way, SG, you want Hillary to win six tosses (because Bernie is unelectable). You toss the first time - Bernie has a 50% chance of winning. Now he wins. What happens? The experiment is over as there is ZERO chance that Hillary will win all six as Bernie has already won one. If Hillary wins the first, and wins the second, third, fourth and fifth as well, the chances for Bernie winning the last one are again 50%, not more, not less.

 

[wareyin]

The article says Sanders won 6 of 7 coin flips (5-1 against Clinton, 1-0 against O'Malley) that CNN could verify through some MS app but admitted that nearly half of the precincts weren't using the app that they could check records on and thus didn't know how many coin flips occurred within them or what the results of them might have been. And then it goes into how the coin flips are for county delegates, not state delegates, and how the flips should have had minimal impact on the state results.

Which, again, is contrary to the claims of Clinton winning every toss.

 

[catsmate]

Clinton is most likely going to be forced out from this email scandal and you think a good ol fashion primary fight is turmoil? I can't wait to see your reaction when all this **** lands.

Hillary Clinton Is Not Electable

Well? Is Hillary done yet?

The fringe will be saying them when she's leaving office.

 

[The Big Dog]

The fringe will be saying them when she's leaving office.

would that be her office at Goldman Sachs? I mean, I assume she has one there given how much time she spent there in 2013.

Her acceptance of $675,000 for three speeches made to the investment firm in 2013 has been a major sticking point for Sanders throughout the debates, and Clinton potentially being beholden to Wall Street one of her biggest criticisms.

“I made speeches to lots of groups.” And when he pressed her on the hundreds of thousands of dollars she accepted from the firm, Clinton offered perhaps the worst answer yet:

Well, I don’t know. That’s what they offered. Every Secretary of State I know has done that.

I disagree, I still think blaming 9/11 as the reason Hillary took all that Wall Street money is the worst excuse ever.

 

[Grizzly Bear]

By losing to Clinton you are winning?

Hillary was blowing the competition out a year ago in the polls. On one hand she won the caucus... but on the other that notion that she had no serious competition in this case isn't helped either. Sanders didnt lose by much at all. So sanders supporters can view that as a relative success

 

[PhantomWolf]

Hillary was blowing the competition out a year ago in the polls. On one hand she won the caucus... but on the other that notion that she had no serious competition in this case isn't helped either. Sanders didnt lose by much at all. So sanders supporters can view that as a relative success

The thing is that a year ago there was no opposition to Hillary because no-one knew anyone else. To expect her numbers to remain static as other players made themselves known was crazy. In the end however, Iowa was one of the States that is strong in the democratic demographic that is showing major support for Bernie, white under-45 males, and yet he tied it. Commentators that looked at the demographics of the State were saying he could very likely win, and on top of that, the numbers that they saw with Obama didn't play out either. Bernie's supporters are clinging to threads of hope, but the reality is that unless something chances dramatically in the next few weeks, after NH, it's going to be all uphill for him.

 

[Trakar]

The thing is that a year ago there was no opposition to Hillary because no-one knew anyone else. To expect her numbers to remain static as other players made themselves known was crazy. In the end however, Iowa was one of the States that is strong in the democratic demographic that is showing major support for Bernie, white under-45 males, and yet he tied it. Commentators that looked at the demographics of the State were saying he could very likely win, and on top of that, the numbers that they saw with Obama didn't play out either. Bernie's supporters are clinging to threads of hope, but the reality is that unless something chances dramatically in the next few weeks, after NH, it's going to be all uphill for him.

It's been all uphill for him for the last 8 months, I don't think anyone, including Sanders, ever thought things were going to become easier. He's gained a lot of ground, but there's still a lot more ground to gain before either primary or national elections are over, for either Democratic candidate.

 

[Skeptic Ginger]

Here is the thing: we don't care about the order. Each sequence is equally probable, but some sequences represent the same outcome. And not all outcomes have the same amount of possible sequences.

I probably wasn't very clear. Let me try again.

People are looking at, "what is probability of 6 heads?" as if that is the only possible question. It isn't the only possible question.

If you are only talking about probability of 6 heads, I don't disagree at all.

But if instead of asking, what is the probability we ask, "how unusual is it to get 6 heads in a row?" it is a different question. And the answer is, it's not unusual at all.

One is theoretical probability, which cannot be applied to the realty of 6 actual coin tosses. You will not get 3 heads and 3 tails every time. Because probability explains the odds of large samples.

Toss a coin 100 times and record the result. Is it unusual (or odd) to get 6 heads in a row among those 100 coin tosses? I will assume we agree it would not be unusual.

Given that a run of 6 is not unusual in a 100 coin toss, is it more or less likely that run of 6 will happen with the first 6 tosses than somewhere in the middle? There is nothing inherent in the coin toss that says you have to toss the coin more than six times to get 6 heads in a row.

Every coin toss has a 50:50 chance of heads or tails.

If you forget about heads and tails for a minute and assign each combination a name. Of your 64 names what is the probability of any one name coming up?

Or try this, if you toss a coin and you get 5 heads in a row, are you now more likely to get tails in the next toss?

Probability is one thing. And we agree on the probability of 6 heads in 6 tosses. So what then explains the fact one gets less probable outcomes? Unless you are talking about a huge number of tosses, the probability of getting 6 heads isn't useful. Especially not in this case.


It doesn't mean I don't agree with you. It means I think one can look at the probability differently than just the probability of 6 heads.

Given there are 64 different possible combinations (remember this is 6 coin tosses here, not 6,000 tosses) it's more useful to consider each of those combinations has an equal chance of occurring.

 

[Skeptic Ginger]

So, what are the odds that Gowdy's committee would leak six sets of email in a row? (Apparently, like these six coin tosses, it's now 1:1.)

 

[Foolmewunz]

More important, of course, is that the six-out-of-six-how'd-that-happen didn't happen. It's the product of an election night rumor, picked up and run with by the Sanders supporters, who should surely take a look at themselves and note the similarities between this sort of conspiracy theory mongering and the Ronulans of previous elections.

 

[Skeptic Ginger]

Each precinct delegate is worth approximately 1/8th of a state-wide convention delegate, and (statistically) an even smaller fraction of a national convention delegate. So each coin toss, which was at most deciding one precinct delegate,

Wrong. The coin tosses decided a caucus outcome, not a precinct outcome.

Given precinct delegates were fixed based on the 11,065 (or whatever the number was), caucus delegates are not likely to result in ties among precinct totals.

So you got the conclusion right. The coin tosses decided a tiny fraction of the 11,065 caucus delegates which will then end up deciding the 1,681 that eventually decides 44 state delegates. (There are other delegates from Iowa not decided by the caucuses.)