On a math ed listserve (purposely not identified), ES asked the following question:
SUBJECT: Odds & Chances
If a company had 39 employees and they laid off the 3 oldest employees (in chronological order from oldest to youngest), does anyone know what the odds of that occuring by chance would be?
How would you respond?
Note: In the Solution Commentary available in a subsequent week, responses submitted via the listserve will be included...some right...and some very wrong.
Hint: Can you simulate the problem? Does order matter? Do odds and probability mean the ssame thing?
Solution Commentary: Consider these suggested solutions (right, wrong, and vague)...try to determine if your solution is right or wrong...or vague:
Perhaps you might look at it this way: First compute how many ways you can choose a group of three employees out of 39. One over that will be the probability of picking a particular group. Now determine how many ways--with respect to choosing (i.e. your first pick, etc.) -- that you could have a group of the three oldest employees and multiply that by the probability of picking a particular group (this amounts to adding up the specified groups since they were disjoint events)...
Looks like 3 in 39 (or 1 in 13)--close to significant.
In chronological order, wouldn't the probability be one out of 39 for laying off the oldest employee, 1 out of 38 for the next oldest from the now pool of 38 workers, then 1 out of 37, so for the sequential events, (1/39)x(1/38)x(1/37) = 1 out of 54,834. As for odds, I don't remember if you multiply them like you do probabilities. My inclination is that you look at the final probability and say the odds are 1 to laying off the three oldest in chronological order to 54,833 to not laying off the oldest. In any event, the layoffs in that order were probably not selected by chance--although any three selections would have the same probability.
The same as the chance of laying off the youngest--or any 3 employees previously specified.
1. The product of (3/39)(2/38)(1/37) = 0.0108% (1 chance in 10,000) without taking into account order.
2. The product of (1/39)(1/38)(1/37) = 0.0018% (2 chance in 100,000) taking into account order.
FROM: MH (again)
In this case, probability and odds are essentially the same because of the relative size of the two numbers. However, probability is the ratio of number possibilities FOR an event to TOTAL possibilities. An odds ratio compares the number of possibilities FOR an event to the number of possibilities AGAINST the evnt. For example, if the probability of "winning" is 1 to 4, the odds for winning are 1 win to 3 losses.