Using 23 children in the same class you calculate the probability of none have the same birthday and then subtract that from 1.

See here for example : http://en.wikipedia.org/wiki/Birthday_problem

This looks a lot simpler to understand : http://www.agenarisk.com/resources/probability_puzzles/birthdays.shtml

From memory I first discovered this in one of Martin Gardner's articles back in the late sixties / early seventies. Either that or a book on reasonably basic probability.

Good idea of yours to post - gets the old grey cells working.

When do the odds approach 99% 95% 90% 80% 70% ETC?

How would one calculate THOSE kinds of odds without doing iterative calculations?

Is there a formula to start with plugging in 95% 90% 80% etc. as the desired outcome and figuring out what the sample size needs to be? I'm sure there is. I'm sure I might have learned this in statistics and probability in college, but I forgot how to do that. Does it have a name? How can I search for the formula or theorem?

EDIT TO ADD: OH SHIT... your graph has all the answers!!!! "BIRTHDAY PARADOX"

If I ever were teaching a high school math class, or substituting in one, I would use this paradox to start the class.

that births can be seasonal which is mentioned in at least one of those links.

My knowledge is very very rusty other than remembering the subject of your OP itself and knowing where to look for appropriate links.

I also recall being at the bar at the local rugby club and finding out that one it was the birthday of one of the youngsters on my own birthday. Lo and behold another one of the players said it was his too and it just so happened that when we were at school we were both in same class. QED..............

fully conforming to human behaviors, there are more births per million in certain months in the USA, and more births per million in other nations, (Australia, New Zealand, South Africa, etc) because of seasonal differences. There are more births per million in certain months in predominantly Muslim or Jewish or Hindu or Buddhist nations than in predominantly Christian nations. Fewer births in some winter in far northern regions, (Sweeden, Norway, etc) than in some summer months. This has to do with the likelihood, (probability) of two people of the opposite sex getting together 9 months earlier to procreate.

Bottom line: the mathematical model is smoothed out to avoid these natural variances due to human customs and habits, or due to seasonal variations of weather. But given a region of similar religious and cultural affiliation in the world, (say Denmark, or Quebec, where there's winter weather about an equal number of nights a year, and where both nations are following a similar religious custom [Christmas, etc.]... the likelihood of births to agglomerate into similar date patterns is reasonable, with occasional outliers more reasonable in those areas than if Denmark is compared to Saudi-Arabia, where both weather and cultural customs are markedly different all year long.

The mathematical model is more statistically accurate when applied to the larger sampling of all 7 billion people on the planet than it is to any one nation, region, climactic and/or cultural area, where normal variances are predicted by one of those latter factors.

apply these same paradoxes to health care, to life insurance, or other ways the corporate world fools us into thinking chances are low, when they really are high, or vice versa?

premiums are assessed by actuaries - the probability of death under given current circumstances such as current age , health history geographical location etc.

I would not endevour to explain your health care premiums over there . Suffice it to say that ignoring our NHS over here which covers everything from the time of conception though to pushing up the daisies , there is no such thing as private cover for pre-exisiting conditions other than maybe at higher rates and that has been so since the '70s. Private health insurance premiums here also increase with age, quite substantially , which I why I dumped that lark about 10 years ago.

btw - I would doubt I am a smarter person that you. Were that to be the case I wouldn't have 80 or so banjos and at least 25 guitars none of which I can play.

Last edited Mon Feb 20, 2012, 09:15 PM - Edit history (1)

the multiple variables of health care coverage in the USA, and not well-informed about probability or statistics.

So the 100+ million UK folks over there have better coverage, more preventive care, all sorts of other variables than those of us in the USA. Most of the USA, probably 89-90% NOW covered, not close to 100% but better than before Obama.

Sorry, too many variables, and too many differences for us to look at populations with which we are familiar. Let's just say, "vive la difference"

Now, on to my theory: that your "birthday paradox" graph looks a lot like an inverse of that graph. The inverse sA hows us more people dying early, a steady number until few die, (because only a a few are left) at the end of the graph. The important part is when the graph looks pretty steady, where about each and every year adds an equal amount of people to the likelihood of death. There's where the premiums must be highest, and the payouts lowest.

To be sure, some idle banker in the 18th century came up with this as he sat with his basic knowledge of probability and statistics, and had nothing else to do but bet which old person walking into the bank would be alive the next year. A thousand re-alignments of that graph later, insurance companies know when to sell policies to 55 year olds, when to stop selling or renewing them to 85 year olds....

This is how modern insurance industries are structured.

You sell 10,000 policies to 10,000 55 year old's, promising 1,000 to every policy holder per premium of $20. that's $2,000,000. You know that, out of an average of 10,000 55-year olds, only 40 people will die, and collect their $1000 in benefits. That's 2
00,000...............profit: 90% of premiums plus premiums, about 8% of premiums, or 2% of premiums collected actually paid out. 98% reciepts over payouts, better than any other merchandising where you have to buy (pay), on average, 50% of what you collect in sales.

...and more than 90% of men will suffer some form of actionable sexual assault in their lifetimes.

Rather frightening isn't it?

Although your statistic might be correct, (I don't know), but the mathematical method would not apply.

The birthday paradox method describes a randomness of (one in 365),(one in 364) etc.

Actionable sexual assault can be committed by one party multiple times, so there is no one in 365 formula to apply.

For the purpose of this exercise the number of offenders and their profligracy is irrelevant.

What matters is how many assaults per 100,000 (or however many) population occur per annum.

And I did neglect the proviso "at least once".

Most (but not all) sexual assaults are a one time occurrence in a victim's life. However the offender is more often than not a multiple offender, and may go on without ever being apprehended, and the only thing to stop them is either prison or death, and not all offenders go to prison, nor do they necessarily stop offending once released from a prison sentence. Some offenders are previous victims, and some have never been previous victims.

So the nature of randomness is much different than one in 365, etc.

... "anybody born on New Years Day."

Three people raised their hands.