A certain town is served by two hospitals. In the larger hospital, about 45 babies are born each day, and in the smaller hospital, about 15 babies are born each day. As you know, about 50 percent of all the babies are boys. However, the exact percentage varies from day to day. Sometimes it may be higher than 50 percent, sometime lower. For a period of one year, each hospital recorded the days in which more than 60 percent of the babies were boys. Which hospital do you think recorded more such days.
Inductive reasoning
Inductive reasoning
I am taking a negotiation course this semester and in the book there is a nifty little exercise on inductive reasoning. I am curious to see if the folks here at the DBB can duplicate the success rates in answering the question that the book states. Read the question then answer using the poll.
That was the way I reasoned it. Flip a coin 10 times, and you could have 7 heads and 3 tails. Flip it 100 times, and you'll most likely have a result much closer to 50-50 than 70-30.Sirius wrote:Inductive reasoning, or just statistical probability? The smaller the sample, the greater the standard deviation...
I don't see how this applies to inductive reasoning. There are two types of that: mathematical and human.
Mathematical induction is your general "prove theorem holds for all in greater than some base case" by proving for base case, and case k+1. Most people know about this one.
Human induction is pretty much the exact opposite of statistical induction with some innate probability assigned to it. I.e. "the lights are on, therefore the lightswitch is in the on position", wherein the lightswitch is really the object controlling the lights, and not the other way around. Interestingly enough even though this mode of thinking is completely unacceptable from a mathematical and scientific point of view, it's the one we use most often. "The car is stopped so it must be out of gas" is another. In logic terms: s->t. See t, assume s with some degree of probability.
Mathematical induction is your general "prove theorem holds for all in greater than some base case" by proving for base case, and case k+1. Most people know about this one.
Human induction is pretty much the exact opposite of statistical induction with some innate probability assigned to it. I.e. "the lights are on, therefore the lightswitch is in the on position", wherein the lightswitch is really the object controlling the lights, and not the other way around. Interestingly enough even though this mode of thinking is completely unacceptable from a mathematical and scientific point of view, it's the one we use most often. "The car is stopped so it must be out of gas" is another. In logic terms: s->t. See t, assume s with some degree of probability.
50%. It'll either be 100% boys or 100% girls on any given day, and it will average out to 50% of each.Drakona wrote:An extreme example illustrates the principle:
Suppose one of the hospitals was so small it only had one baby born per day. Over the course of a year, how many days would you expect the hospital to deliver at least 60% boys on that day?
Obviously. The point is that given n days you'd breach the threshold on a given day n/2 times (on average) with one baby born. With 2 it would be n/4 (4 options).DCrazy wrote:50%. It'll either be 100% boys or 100% girls on any given day, and it will average out to 50% of each.
Actually now that I think about it I'd be inclined to vote the non-existant 3rd option. It seems like this problem over a certain number of babies born (that's smaller than one would initially think, at least without strong statistical background) it would be statistically similar.
Hmm. Perhaps an everyday example would illustrate the principle better.
When I play Descent (or when I did...
), I usually had an idea of how I would do against specific pilots, long-term. For example, when I played Soulvoid, I expected to go 3-5 over the course of the game. That's a long term ratio. Depending on who I was playing, it'd be anywhere from 1-3 to 5-1.
Over very short games, (say, 2 kills), how many did I break 80%? That actually happened pretty regularly. Sometimes I'd just get the first two kills.
Over short games (say, 5-10 kills), how often did I break 80%? That was rare, but it happened. Sometimes a game would open up 5-1, and it didn't mean much... I was just doing well coming into the game.
Over medium games (say, 20 kills), how often did I break 80%? Almost never. If I did, either my opponent was having a terribly off day, or I was fighting a n00b, or something else was up.
Over long games (say, 50-100 kills), how often did I break 80%? Never. At least, never when it was unexpected.
When I play Descent (or when I did...
), I usually had an idea of how I would do against specific pilots, long-term. For example, when I played Soulvoid, I expected to go 3-5 over the course of the game. That's a long term ratio. Depending on who I was playing, it'd be anywhere from 1-3 to 5-1. Over very short games, (say, 2 kills), how many did I break 80%? That actually happened pretty regularly. Sometimes I'd just get the first two kills.
Over short games (say, 5-10 kills), how often did I break 80%? That was rare, but it happened. Sometimes a game would open up 5-1, and it didn't mean much... I was just doing well coming into the game.
Over medium games (say, 20 kills), how often did I break 80%? Almost never. If I did, either my opponent was having a terribly off day, or I was fighting a n00b, or something else was up.
Over long games (say, 50-100 kills), how often did I break 80%? Never. At least, never when it was unexpected.
Well, you DBB'ers are smater than the average bear. Here is the answer straight out of the book.
Inductive reasoning is a form of hypothesis testing, or trial and error. In general, people are not especially good at testing hypotheses, and they tend to use confirmatory methods. An example is the availability heuristic, such that judgments of frequency tend to be biased by the ease with which information can be called to mind.
For example, people make inaccurate judgments when estimating probabilities. Consider the hospital problem. When people are asked to answer this question, 22 percent select the first answer (i.e., the larger hospital), 22 percent select the second answer (i.e., the smaller hospital), and 56 percent select the third answer (i.e., both hospitals). They seem to make no compensation for large versus small sample sizes. They believe that an extreme event â?? for example, 60 percent of births being male â?? is just as likely in a large hospital as in a small one. In fact, it is actually far more likely for an extreme event to occur within a small sample because fewer cases are included in the average. People often fail to take sample size into account when they make an inference.
the question wasn't if an extreme event would happen or not. it was if one of a total possible 2 extreme events would happen.
no matter how small the sample, the probabaility is 50/50 for either 40% boys born, or 60% boys born. it's equal either way.
my point is that there is no greater possability for there being a 60% boys event, over a 40% boys event.
i restate, i agree that it would be more likely that an extreme event would happen in the smaller sample, however it's not more likely one extreme event would happen over another (when there are multiple possible extremes).
.oh!...
scratch all that, i just re-read the question a few times. i just saw the crux of the matter.
it's testing the number of 60% extreme events, (which would be just as high as the number of 40% extreme events, but the ratio does not matter, only the raw number matters). they are not asking for a ratio output from each respective hospital itself (which would allow both extremes to cancel eachother out).
i got it wrong coz i misread it.
no matter how small the sample, the probabaility is 50/50 for either 40% boys born, or 60% boys born. it's equal either way.
my point is that there is no greater possability for there being a 60% boys event, over a 40% boys event.
i restate, i agree that it would be more likely that an extreme event would happen in the smaller sample, however it's not more likely one extreme event would happen over another (when there are multiple possible extremes).
.oh!...
scratch all that, i just re-read the question a few times. i just saw the crux of the matter.
it's testing the number of 60% extreme events, (which would be just as high as the number of 40% extreme events, but the ratio does not matter, only the raw number matters). they are not asking for a ratio output from each respective hospital itself (which would allow both extremes to cancel eachother out).
i got it wrong coz i misread it.