# Why Does the Probability of Each Trial Approach Zero in a Continuum?

• eterna
In summary: The probability of a particular number of successes may increase or decrease depending on the situation.In summary, the probability of success for each trial in a Poisson process approaches zero as the number of trials increases, but it is not fixed.
eterna

## Homework Statement

From a site introducing the Poisson Distribution

"When trials can occur in a fixed continuum of time (or distance), each instant of time (or distance) is essentially a distinct trial. Because a continuum contains an infinity of points, this means a statistical experiment may have an infinite number of trials, and the probability of each trial would approach zero."

can someone explain why the probability of success for each trial would approach zero? Isn't it fixed?

## The Attempt at a Solution

The total probability of the distribution cannot be bigger than 1.

Simon Bridge said:
The total probability of the distribution cannot be bigger than 1.

so that means the probability of getting a specific number of successes is almost zero since there are an infinite amount of trials?

As the number of chances for a success increases - the mean number of successes increases.
The probability of a particular number of successes may increase or decrease depending - i.e. if there are fewer than 10 trials then the probability of 11 or more successes is zero - increase the number of trials and P(x>10) increases. P(x=10) initially increases and then decreases as the number of trials increases.

The probability of the most likely value decreases as the number of trials increases.

You should plot Poisson distributions for several situations to see how it behaves.

eterna said:

## Homework Statement

From a site introducing the Poisson Distribution

"When trials can occur in a fixed continuum of time (or distance), each instant of time (or distance) is essentially a distinct trial. Because a continuum contains an infinity of points, this means a statistical experiment may have an infinite number of trials, and the probability of each trial would approach zero."

can someone explain why the probability of success for each trial would approach zero? Isn't it fixed?

## The Attempt at a Solution

Imagine a process in which events can occur at any time, but whose average occurrence is 24 events per day. Imagine also that the inter-event times are independent and identically distributed, so whether an event occurs right now will not affect the probability that another event will, or will not occur in the next second, or next two seconds, etc. The result will be a Poisson process, in which the distribution of the number of events in a time interval will be a Poisson random variable whose mean is proportional to the interval's length (so it we double the length of the interval we double the expected number of events occurring in the interval). So, we have on average, 1 event per hour, or "1/60 of an event per minute", which really means that we have a probability of about 1/60 that an event will occur in any specified 1-minute interval. We have "1/3600 of an event per second", meaning that in any given second there is a probability of about 1/3600 that an event will occur. In a microsecond there is a probability of ##10^{-6}/3600## that an event will occur, etc. So, as the time interval shrinks to zero, the probability of an event occurring therein shrinks to zero as well. Nevertheless, since a finite time interval consists of a huge number of tiny intervals, the probability of occurrence is made up from a huge number of tiny probabilities, and so can amount to a sizable value.

The original statement is badly phrased. It is NOT correct that " the probability of each trial would approach zero." The probability of "each trial" is zero. However, in a continuous situation like this you would never ask about individual trials, you would only ask about intervals.

HallsofIvy said:
The original statement is badly phrased. It is NOT correct that " the probability of each trial would approach zero." The probability of "each trial" is zero.
Well: strictly speaking, the statement "the probability of each trial" is nonsense - you can have a probability of a particular "outcome" of each trial. Each trial may have many outcomes, each with their own probability.

I concur.

In a continuous distribution, the probability of an outcome would depend on an interval of the probability density function for that outcome.

eterna said:
From a site introducing the Poisson Distribution
... we really need a link to the site in question - it could just be that the site is rubbish.
... looks like it may be this one:
http://www.milefoot.com/math/stat/pdfd-poisson.htm 

The question revisited:
...can someone explain why the probability of success for each trial would approach zero? Isn't it fixed?
... short answer: no - it is not fixed.
But context is everything.
 using the above website: the quoted passage does not refer to the poisson distribution - the second half of the passage spells out that the rate of successes is a constant, but the probability of a particular number of successes in an interval is not.

The passage being considered is:
"When trials can occur in a fixed continuum of time (or distance), each instant of time (or distance) is essentially a distinct trial. Because a continuum contains an infinity of points, this means a statistical experiment may have an infinite number of trials, and the probability of each trial would approach zero."

I don't think the passage quoted is actually supposed to refer to any special property of the Poisson distribution.
I think the author is trying to make a connection between discrete and continuous probabilities.

Consider:
Say I roll a die every minute:
If N is the number of sixes in an hour, then ##P(N=n)\sim \text{Pois}(n;10)##: $$p(n)=\frac{10^ne^{-10}}{n!}$$
A single trial would, therefore, be a single hour-long observation.

The author seems to be saying that if the trial interval is shorter, then p(n) gets smaller.

If I make the trial interval T shorter, then $$p(n;T)=\frac{(T/6)^ne^{-T/6}}{n!}$$... if T is in minutes.

... the effect is to make small values of n more likely as T decreases: how many sixes would I expect in 10 seconds, 5, 1?

Imagine we define a "success" as the event that N=0 (i.e. no sixes are rolled in the time interval)
Then the probability of a success, ##p(0;T)=e^{-T/6}##, approaches 1 and T approaches 0.

... this seems at odds with the passage quoted - which is why I suspect this passage must come before the Poisson distribution is actually introduced. Either that or the author got confused.

Certainly the statement requires a lot of qualification to make it work sensibly.
So how about it eterna: any of this help?

... if the link is as above, then the whole passage goes like this:
When trials can occur in a fixed continuum of time (or distance), each instant of time (or distance) is essentially a distinct trial. Because a continuum contains an infinity of points, this means a statistical experiment may have an infinite number of trials, and the probability of each trial would approach zero. But the rate of successes per unit of time (or distance) can still be a finite, nonzero quantity.

Let us assume that there are only two possible outcomes for each instant of time (or distance), that two successes cannot occur at the same instant, that trials are independent, and that the rate of successes is constant. Let X represent the number of successes. Then X has a Poisson distribution.​
... you left out the bit not in italics.
I also separated out the passage into two paragraphs to help clarity.
The first paragraph is more general concerning continuous distributions and the second attempts to motivate the Poisson distribution.

Last edited:

## What is simple probability?

Simple probability is a way of measuring the likelihood of an event occurring. It is the ratio of the number of favorable outcomes to the total number of possible outcomes.

## How is simple probability calculated?

Simple probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. This is typically represented as a fraction or decimal.

## What is the difference between theoretical and experimental probability?

Theoretical probability is based on the assumption that all outcomes are equally likely, while experimental probability is based on actual data collected from experiments or observations.

## What is the probability of an event with certain and impossible outcomes?

An event with certain outcomes has a probability of 1, meaning it is guaranteed to occur. An event with impossible outcomes has a probability of 0, meaning it cannot occur.

## How can simple probability be applied in real life?

Simple probability can be applied in various real-life situations, such as predicting the outcome of a coin toss, estimating the chances of winning a game, or determining the likelihood of rain on a given day.

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