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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
In the simplest case, if one allocates balls into bins (with =) sequentially one by one, and for each ball one chooses random bins at each step and then allocates the ball into the least loaded of the selected bins (ties broken arbitrarily), then with high probability the maximum load is: [8]
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 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?
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 ...
Many probability text books and articles in the field of probability theory derive the conditional probability solution through a formal application of Bayes' theorem — among them books by Gill [51] and Henze. [52] Use of the odds form of Bayes' theorem, often called Bayes' rule, makes such a derivation more transparent. [34] [53]
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.
In this basic urn model in probability theory, the urn contains x white and y black balls, well-mixed together. One ball is drawn randomly from the urn and its color observed; it is then placed back in the urn (or not), and the selection process is repeated. [3] Possible questions that can be answered in this model are: