# 数学代写|概率论代写Probability theory代考|INFINITE SEQUENCES OF TRIALS

#### Doug I. Jones

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## 数学代写|概率论代写Probability theory代考|INFINITE SEQUENCES OF TRIALS

In the preceding chapter we have dealt with probabilities connected with $n$ Bernoulli trials and have studied their asymptotic behavior as $n \rightarrow \infty$. We turn now to a more general type of problem where the events themselves cannot be defined in a finite sample space.

Example. A problem in runs. Let $\alpha$ and $\beta$ be positive integers, and consider a potentially unlimited sequence of Bernoulli trials, such as tossing a coin or throwing dice. Suppose that Paul bets Peter that a run of $\alpha$ consecutive successes will occur before a run of $\beta$ consecutive failures. It has an intuitive meaning to speak of the event that Paul wins, but it must be remembered that in the mathematical theory the term event stands for “aggregate of sample points” and is meaningless unless an appropriate sample space has been defined. The model of a finite number of trials is insufficient for our present purpose, but the difficulty is solved by a simple passage to the limit. In $n$ trials Peter wins or loses, or the game remains undecided. Let the corresponding probabilities be $x_n, y_n, z_n$ $\left(x_n+y_n+z_n=1\right)$. As the number $n$ of trials increases, the probability $z_n$ of a tie can only decrease, and both $x_n$ and $y_n$ necessarily increase. Hence $x=\lim x_n, y=\lim y_n$, and $z=\lim z_n$ exist. Nobody would hesitate to call them the probabilities of Peter’s ultimate gain or loss or of a tie. However, the corresponding three events are defined only in the sample space of infinite sequences of trials, and this space is not discrete.
The example was introduced for illustration only, and the numerical values of $x_n, y_n, z_n$ are not our immediate concern. We shall return to their calculation in example XIII, (8.b). The limits $x, y, z$ may be obtained by a simpler method which is applicable to more general cases. We indicate it here because of its importance and intrinsic interest.

Let $A$ be the event that $a$ run of $\alpha$ consecutive successes occurs before a run of $\beta$ consecutive failures. In the event $A$ Paul wins and $x=\mathbf{P}{A}$. If $u$ and $v$ are the conditional probabilities of $A$ under the hypotheses, respectively, that the first trial results in success or failure, then $x=p u+q v$ [see V, (1.8)]. Suppose first that the first trial results in success. In this case the event $A$ can occur in $\alpha$ mutually exclusive ways: (1) The following $\alpha-1$ trials result in successes; the probability for this is $p^{\alpha-1}$. (2) The first failure occurs at the $v$ th trial where $2 \leq v \leq \alpha$. Let this event be $H_v$. Then $\mathbf{P}\left{H_v\right}=p^{v-2} q$, and $\mathbf{P}\left{A \mid H_v\right}=v$. Hence (using once more the formula for compound probabilities)
$$u=p^{\alpha-1}+q v\left(1+p+\cdots+p^{\alpha-2}\right)=p^{\alpha-1}+v\left(1-p^{\alpha-1}\right) .$$
If the first trial results in failure, a similar argument leads to
$$v=p u\left(1+q+\cdots+q^{\beta-2}\right)=u\left(1-q^{\beta-1}\right) .$$
We have thus two equations for the two unknowns $u$ and $v$ and find for $x=p u+q v$
$$x=p^{\alpha-1} \frac{1-q^\beta}{p^{\alpha-1}+q^{\beta-1}-p^{\alpha-1} q^{\beta-1}} .$$
To obtain $y$ we have only to interchange $p$ and $q$, and $\alpha$ and $\beta$. Thus
$$y=q^{\beta-1} \frac{1-p^\alpha}{p^{\alpha-1}+q^{\beta-1}-p^{\alpha-1} q^{\beta-1}} .$$

## 数学代写|概率论代写Probability theory代考|SYSTEMS OF· GAMBLING

The painful experience of many gamblers has taught us the lesson that no system of betting is successful in improving the gambler’s chances. If the theory of probability is true to life, this experience must correspond to a provable statement.

For orientation let us consider a potentially unlimited sequence of Bernoulli trials and suppose that at each trial the bettor has the free choice of whether or not to bet. A “system” consists in fixed rules selecting those trials on which the player is to bet. For example, the bettor may make up his mind to bet at every seventh trial or to wait as long as necessary for seven heads to occur between two bets. He may bet only following a head run of length 13, or bet for the first time after the first head, for the second time after the first run of two consecutive heads, and generally, for the $k$ th time, just after $k$ heads have appeared in succession. In the latter case he would bet less and less frequently. We need not consider the stakes at the individual trials; we want to show that no “system”‘ changes the bettor’s situation and that he can achicve the same result by betting every time. It goes without saying that this statement can be proved only for systems in the ordinary meaning where the bettor does not know the future (the existence or non-existence of genuine prescience is not our concern). It must also be admitted that the rule “go home after losing three times”‘ does change the situation, but we shall rule out such uninteresting systems.
We define a system as a set of fixed rules which for every trial uniquely determine whether or not the bettor is to bet; at the kth trial the decision may depend on the outcomes of the first $k-1$ trials, but not on the outcome of trials number $k, k+1, k+2, \ldots$; finally the rules must be such as to ensure an indefinite continuation of the game. Since the set of rules is fixed, the event “in $n$ trials the bettor bets more than $r$ times” is well defined and its probability calculable. The last condition requires that for every $r$, as $n \rightarrow \infty$, this probability tends to 1 .

# 概率论代考

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