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By Robert J. Adler

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Extra resources for Introduction to continuity, extrema and related topics for general Gaussian processes

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The wheel is spun and players wager that a ball will fall into one of the partitions. If the ball lands on the selected number (or one of a set of selected numbers) the player wins; otherwise the player loses. For example, say the player places a $1 bet on number 7. If the ball lands on this number, the player wins $35. If not, the bet is lost. The probability of winning this bet is 1/38 and the player’s expectation for a single bet of this type is E= 1 37 1 ($35) + (−$1) = −$ ≈ −5¢. 38 38 19 By betting on any single number, as above, the player loses, on average, approximately 5¢ on each $1 wagered, or 5 percent of the bet.

The “fortunately-unfortunately” tale below shows an increased and decreased likelihood of a happy ending as additional information becomes available: There was a man flying up in the sky. Fortunately he was in an airplane. Unfortunately he fell out of the plane. Fortunately he had on a parachute. Unfortunately the parachute didn’t open. Fortunately there was a haystack below. Unfortunately there was a pitchfork in the haystack. Fortunately he missed the pitchfork. Unfortunately he missed the haystack!

Such faulty reasoning would suggest your probability of living to see tomorrow is 1/2 because there are two possibilities and survival corresponds to one of the two. Is there life on Mars? Would you argue there is a 50 percent chance of this because there are two possibilities? Hopefully not! The complement of event A, denoted as Ā, is the set of all sample points in the sample space not belonging to A. In other words, Ā is the event that A does not occur. If a single die is rolled and A = {2,4,6}, then Ā = {1,3,5}.

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