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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 ...
The primary difference between a computer algebra system and a traditional calculator is the ability to deal with equations symbolically rather than numerically. The precise uses and capabilities of these systems differ greatly from one system to another, yet their purpose remains the same: manipulation of symbolic equations.
One important drawback for applications of the solution of the classical secretary problem is that the number of applicants must be known in advance, which is rarely the case. One way to overcome this problem is to suppose that the number of applicants is a random variable N {\displaystyle N} with a known distribution of P ( N = k ) k = 1 , 2 ...
The Newton–Pepys problem is a probability problem concerning the probability of throwing sixes from a certain number of dice. [1] In 1693 Samuel Pepys and Isaac Newton corresponded over a problem posed to Pepys by a school teacher named John Smith. [2] The problem was: Which of the following three propositions has the greatest chance of success?
The odds strategy is the rule to observe the events one after the other and to stop on the first interesting event from index s onwards (if any), where s is the stopping threshold of output a. The importance of the odds strategy, and hence of the odds algorithm, lies in the following odds theorem.
Suppose l > t.In this case, integrating the joint probability density function, we obtain: = = (), where m(θ) is the minimum between l / 2 sinθ and t / 2 .. Thus, performing the above integration, we see that, when l > t, the probability that the needle will cross at least one line is
One may run this algorithm multiple times returning a false answer if it reaches a false response within k iterations, and otherwise returning true. Thus, if the number is prime then the answer is always correct, and if the number is composite then the answer is correct with probability at least 1−(1− 1 ⁄ 2) k = 1−2 −k.
It is a hard (and often open) problem to calculate the minimum number of tickets one needs to purchase to guarantee that at least one of these tickets matches at least 2 numbers. In the 5-from-90 lotto, the minimum number of tickets that can guarantee a ticket with at least 2 matches is 100.