Sanderson M. Smith
Cate School Mathematics Dept.
Statistical Probability
Applied to Cate School Data
The Cate School Class of 1997 graduated on June 1, 1997. During the ceremonies, Headmaster Peter Thorp announced that 3 seniors in the 67-member graduation class had birthdays on that specific day.
An interesting question: What is the probability that such an event could happen? The question is statistically answerable if we make the assumption that birthdays are not considered when Cate students are selected during the admission process. Clearly the 67 members of the class of 1997 are not a randomly selected group. However, it is reasonable to assume that the collection of birthdays would represent a sample from a random distribution of birthdays over 365 days. With this assumption, we have a binomial distribution with these probabilities:
Probability (birthday on June 1) = 1/365
Probability (birthday not on June 1) = 364/365
If we now consider a samp!
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le of 67 randomly chosen birthdays, then
Probability (0 birthdays on June 1) = 67C0(364/365)67(1/365)0 = 83.2091%
Probability (1 birthday on June 1) = 67C1(364/365)66(1/365)1 = 15.3160%
Probability (2 birthdays on June 1) = 67C2(364/365)65(1/365)2 = 1.3855%
Probability (3 birthdays on June 1) = 67C3(364/365)64(1/365)3 = 0.0827%
Probability (4 birthdays on June 1) = 67C4(364/365)63(1/365)4 = 0.0001% Probability (at least one birthday on June 1) = 16.7909%
To summarize, if we had 10,000 sets of 67 random birthdays, in 8 of these sets we would expect to find 3 birthdays on June 1.
Does this mean that this event is unlikely? The answer to this question depends upon how one chooses to define "likely" or "unlikely." In everyday usage, a probability of 0.0827% (less than 1/10 of 1%) would probably be considered unlikely. However, let's make a brief comparison to California Super Lotto, where one chooses 6 numbers from the set 1,2,3,...,49,50,51 and!
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wins big (often millions of dollars) if the chosen 6 numbers match 6 numbers randomly selected by the State. The number of possible 6-number combinations is
51C6 =18,009,460
The probability of matching the State's six numbers is thus 1/18,009,460.
To put this in perspective, if you bought 50 Super Lotto tickets a week, you would expect to win the jackpot once every 6,900 years. Another perspective: To have a 50% chance of winning the jackpot, you would have to buy one ticket a day for 34,000 years.
Simple arithmetic shows that the probability that 3 of 67 randomly chosen birthdays will be on June 1 is over 14,000 times the probability of hitting the jackpot in California Super Lotto.
Since birthdays were the original topic of discussion, it is interesting to note that if 23 people are chosen by a random process, the probability that two of them have a common birthday is 50.73%. In a random group of 67 individuals, the probability of a common birthday is 99.84%.!
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"There is no smallest among the small and no largest among the large:
But always something still smaller and something still larger."
-Anaxagoras (ca 450 B.C.)