Note that both x = 0 and x = 1 probabilities are 0.5
Example: Astronomers estimate there are 10 billion planets in the Milky Way galaxy that may support life. How
small might the initial probability be for a 50% chance that at least one planet does have life forms? Very
p = lamda / 10 ^10 = 6.93 * 10 ^ - 11
Tidbit # 2
Throw a six sided die three times. What's the probability the numbers are all different? Well, the first
toss can be any number so we could say it has a p = 6/6. The second number has five chances of
being different from the first, so it has p= 5/6. The third must be different from the first two, so it has
p = 4/6. So p = (6/6)(5/6)(4/6) = 120 / 216 = .5555
All 216 possibilities are shown in the list below. Notice that all six columns have twenty with different
numbers. 20 X 6 = 120 out of 216 which we already found the easy way.
Tidbit # 3 Two Coins
Two coins are tossed. One came up heads. What's the probability the other coin also
came up heads?
The answer is 1/3 since TT has been eliminated leaving HH HT TH
This is a simple example of probabilities changing when information is added. The next
example is not so simple.
Tidbit # 4 Monty Hall
The Monty Hall problem received much attention, and the internet has many articles on the subject.
It's based on a old TV program named "Let's Make a Deal". There are three doors, one of which
hides a expensive automobile. The other two hide booby prizes. When a contestant selects a door,
Monty shows what's behind a different door (always a booby prize). He now asks the contestant
whether or not he/she wishes to switch their selection.
Surprisingly, a strategy of switching the selection wins the prize 2/3 of the time. Not switching
only wins 1/3 of the time. You lose by switching only in the case where the initial pick is the door hiding the prize.
Formal methods of solution will appeal to mathematicians, of course. The Monty Hall problem is so
unintuitive that I find it most convincing and satisfying to think it through using sketches like the
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