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For example, if two fair six-sided dice are thrown to generate two uniformly distributed integers, and , each in the range from 1 to 6, inclusive, the 36 possible ordered pairs of outcomes (,) constitute a sample space of equally likely events. In this case, the above formula applies, such as calculating the probability of a particular sum of ...
Diceball! is a board game in which two players roll dice to simulate a baseball game, one representing the visiting team and the other the home team. Both players use the dice to throw the baseball from the mound to the plate and field the ball on defense. Diceball! was designed to mirror the statistical reality of baseball.
Another way to describe collectively exhaustive events is that their union must cover all the events within the entire sample space. For example, events A and B are said to be collectively exhaustive if = where S is the sample space. Compare this to the concept of a set of mutually exclusive events. In such a set no more than one event can ...
These include whether a specially marked die (called the Mayhem die) has rolled highest, the lowest number rolled, and whether any two dice show the same number. One other commonly used variant of the 6-sided dice roll is the d3, which is a 6-sided die roll, with the result divided by 2. The average result is 2, and the standard deviation is 0.816.
A set of dice is intransitive (or nontransitive) if it contains X>2 dice, X1, X2, and X3... with the property that X1 rolls higher than X2 more than half the time, and X2 rolls higher than X3 etc... more than half the time, but where it is not true that X1 rolls higher than Xn more than half the time.
To see the difference, consider the probability for a certain event in the game. In the above-mentioned dice games, the only thing that matters is the current state of the board. The next state of the board depends on the current state, and the next roll of the dice. It does not depend on how things got to their current state.
These two non-atomic examples are closely related: a sequence (x 1, x 2, ...) ∈ {0,1} ∞ leads to the number 2 −1 x 1 + 2 −2 x 2 + ⋯ ∈ [0,1]. This is not a one-to-one correspondence between {0,1} ∞ and [0,1] however: it is an isomorphism modulo zero , which allows for treating the two probability spaces as two forms of the same ...
The problem of estimating the maximum of a discrete uniform distribution on the integer interval [,] from a sample of k observations is commonly known as the German tank problem, following the practical application of this maximum estimation problem, during World War II, by Allied forces seeking to estimate German tank production.
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