Difference between revisions of "Chance News 15"

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* The banker makes one last offer; the contestant accepts that offer or takes whatever money is in the initially chosen briefcase.
 
* The banker makes one last offer; the contestant accepts that offer or takes whatever money is in the initially chosen briefcase.
  
Thus the banker is always trying to buy out the player.  If he fails the player will end up with the amount in the box that he chose.  
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Thus the banker is always trying to buy out the player.  If he fails the player will end up with the amount in suitcase.  
  
The best way to understand the game is to play it [http://www.nbc.com/Deal_or_No_Deal/game/ here ] on the NBC website. For example suppose we play the web version.
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The best way to understand the game is to play it [http://www.nbc.com/Deal_or_No_Deal/game/ here ] on the NBC website.  
  
We choose briefcase 13 for our briefcase. After each round we can calculate the expected amount in our briefcase since it is just the average of the amounts in the briefcases that have not yet been chosen.  Our first instruction was to choose six briefcases. The amounts in these briefcases are no longer available for future rounds.  Here is the result of our first round
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We did this and here is the result of the first round of play.  We chose briefcase 13 for our briefcase.   Our first instruction was to choose six briefcases.Here is the result of the first round.
  
 
<center>[[Image:deal.jpg|400px|After the first round.]]
 
<center>[[Image:deal.jpg|400px|After the first round.]]
 
</center>
 
</center>
  
The banker has offered us $29.854 to quit.  The expected value for the amount in our briefcase is $125,900 so we would have to be pretty risk adverse to give in at this point.   
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Note that we chose suitcases The banker has offered us $29.854 to quit.  The expected value for the amount in our briefcase is $125,900 so we would have to be pretty risk adverse to give in at this point.   
  
 
We played a game all the way through and found the following expected values and banker's offers:
 
We played a game all the way through and found the following expected values and banker's offers:
  
<table width="50%" border="1">
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<center><table width="50%" border="1">
 
   <tr>  
 
   <tr>  
 
     <td><div align="center">Round</div></td>
 
     <td><div align="center">Round</div></td>
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   </tr>
 
   </tr>
 
</table>
 
</table>
 
+
</center>
to be continued
 

Revision as of 16:15, 8 March 2006

Quotation

Take statistics. Sorry, but you'll find later in life that it's handy to know what a standard deviation is.

David Brooks
New York Times, March 2, 2006

This appears on a list of core knowledge that Brooks says will be sufficient to give you a great education even if you don't make Harvard.

Forsooth

From the February 2006 RSS news we have:

One primary school in East London has a catchment area of 110 metres.

Sunday Telegraph
23 October 2005

More on medical studies that conflict with previous studies

Humans, being what they are, it is only natural that when a study's consequences seem plausible there is no need to look too closely. On the other hand, when the outcomes go against what was expected, a great deal of inspection is called for. This was discussed here.

The Wall Street Journal of February 28, 2006 details possible reasons for why the Women's Health Initiative might have had design flaws leading to "murky results." In summary, the WSJ reported:

  • Calcium/Vitamin D study
    • Message: Supplements don't protect bones or cut risk of colorectal cancer.
    • Problem: Those in placebo group also took supplements in many cases.
  • Low-fat diet study:
    • Message: Doesn't cut risk of breast cancer.
    • Problem: Few met the fat goal.
    • A 22% drop in risk for women who cut fat the most got little emphasis.
  • Hormone study:
    • Message: No benefit, possible increased cancer and heart risk.
    • Problem: Most in study were too old for this to apply to menopausal women.

More generally, according to the WSJ, "Design problems in all of the trials mean the results don't really answer the questions they were supposed to address. And a flawed communications effort led to widespread misinterpretation of results by the news media and public."

In particular, in order to reduce the number of participants for the studies, "more than half [of the women] took part in at least two of them, and more that 5,000 were in all three trials." As might be imagined, "Among problems this posed was simple burnout" which "contributed to compliance problems that plagued all three and hurt the reliability of their results."

Another problem was the difficulty of double blinding for the hormone study since any hot flashes would indicate to the patient (and to her physician) that she was in the placebo arm; to get around this impediment, the vast majority of the women recruited were well past menopause, thus biasing the results against the benefits of hormone replacement.

So where are we after 68,132 female participants, "fifteen years and $725 million later"? More than likely, the Women's Health Initiative study will be in and out of the news for some time to come because of its ambiguity.

Submitted by Paul Alper

The economists analyze the tv show "Deal or No Deal?"

Deal or No Deal? Decision making under risk in a large-payoff game show
Thierry Post, Marijn Van den Assem, Guido Baltussen,Richard Thaler
February 2006

The authors of this papere write:

The popular television game show "Deal or No Deal" offers a uniue opportunity for analyzing decision making under risk: it involves very large and wide-ranging stakes, simple sop-go decisions that reuire minimal skill, knowledge or strategy and near-certaint aboutthe probability distribution.

Here is a nice description of the game from the MS Math in the Media magazine:

  • Twenty-six known amounts of money, ranging from one cent to one million dollars, are (symbolically) randomly placed in 26 numbered, sealed briefcases. The contestant chooses a briefcase. The unknown sum in the briefcase is the contestant's.
  • In the first round of play, the contestant chooses 6 of the remaining 25 briefcases to open. Then the "banker" offers to buy the contestant's briefcase for a sum based on its expected value, given the information now at hand, but tweaked sometimes to make the game more interesting. The contestant can accept ("Deal") or opt to continue play ("No Deal").
  • If the game continues, 5 more briefcases are opened in the second round, another offer is made, and accepted or refused. If the contestant continues to refuse the banker's offers, subsequent rounds open 4, 3, 2, 1, 1, 1, 1 briefcases until only two are left.
  • The banker makes one last offer; the contestant accepts that offer or takes whatever money is in the initially chosen briefcase.

Thus the banker is always trying to buy out the player. If he fails the player will end up with the amount in suitcase.

The best way to understand the game is to play it here on the NBC website.

We did this and here is the result of the first round of play. We chose briefcase 13 for our briefcase. Our first instruction was to choose six briefcases.Here is the result of the first round.

After the first round.

Note that we chose suitcases The banker has offered us $29.854 to quit. The expected value for the amount in our briefcase is $125,900 so we would have to be pretty risk adverse to give in at this point.

We played a game all the way through and found the following expected values and banker's offers:

Round
Epected Value
Offer
1
130,857
30565
2
142,475
37361
3
125,186
54507
4
153,319
57554
5
170,967
67160
6
205,160
108950
7
256,425
176781
8
341,800
317700
9
512,500
367500