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However, if one considers 100 confidence intervals simultaneously, each with 95% coverage probability, the expected number of non-covering intervals is 5. If the intervals are statistically independent from each other, the probability that at least one interval does not contain the population parameter is 99.4%.
From a permutations perspective, let the event A be the probability of finding a group of 23 people without any repeated birthdays. Where the event B is the probability of finding a group of 23 people with at least two people sharing same birthday, P(B) = 1 − P(A).
Thus, by assuring , the probability of making one or more type I errors in the family is controlled at level . A procedure controls the FWER in the weak sense if the FWER control at level α {\displaystyle \alpha \,\!} is guaranteed only when all null hypotheses are true (i.e. when m 0 = m {\displaystyle m_{0}=m} , meaning the "global null ...
A simple way to demonstrate that a switching strategy really does win two out of three times with the standard assumptions is to simulate the game with playing cards. [58] [59] Three cards from an ordinary deck are used to represent the three doors; one 'special' card represents the door with the car and two other cards represent the goat doors.
Specifically, that two different procedures for determining that "at least one is a boy" could lead to the exact same wording of the problem. But they lead to different correct answers: From all families with two children, at least one of whom is a boy, a family is chosen at random. This would yield the answer of 1 / 3 .
In probability theory, the coupon collector's problem refers to mathematical analysis of "collect all coupons and win" contests. It asks the following question: if each box of a given product (e.g., breakfast cereals) contains a coupon, and there are n different types of coupons, what is the probability that more than t boxes need to be bought ...
Then a coupling of and is a new probability space (,,) over which there are two random variables and such that has the same distribution as while has the same distribution as . An interesting case is when Y 1 {\displaystyle Y_{1}} and Y 2 {\displaystyle Y_{2}} are not independent.
Graphs of probability P of not observing independent events each of probability p after n Bernoulli trials vs np for various p.Three examples are shown: Blue curve: Throwing a 6-sided die 6 times gives a 33.5% chance that 6 (or any other given number) never turns up; it can be observed that as n increases, the probability of a 1/n-chance event never appearing after n tries rapidly converges to 0.