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The event {X = x j } is defined as the set of those outcomes for which the random variable X takes on the value x j . The probability of the event {X = x j } is thus defined as the sum of the probabilities of the individual outcomes for which X takes on the value x j . This probability is denoted by P(X = x j ). The function p j = P(X = x j ) for j = 1, . . , M is called the probability mass function of X . The possible values x1 , . . , x M are called mass points of X . 1 John and Mary each roll one die.

A probability measure must satisfy a number of consistency rules that will be discussed later. Let’s first illustrate a few things in light of an experiment that children sometimes use in their games to select one child out of the group. Three children simultaneously present their left or right fist to the group. ” The sample space of this experiment can be described by the set S = {R R R, R R L , R L R, R L L , L L L , L L R, L R L , L R R} consisting of eight elements, where R(L) stands for a right (left) fist.

3. Average payoff in St. Petersburg game. Roulette is one of the oldest casino games. The basis for this game was established at the beginning of the seventeenth century by French mathematician Blaise Pascal. The most common version of roulette uses the numbers 0, 1, . . , 36, where the number 0 is always reserved as winning number for the house (European roulette). 6 Roulette and the law of large numbers 43 when the roulette wheel stops spinning. Bets may be placed on either single numbers or combinations of numbers.

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