Roulette Bernoulli Experiment

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Roulette Bernoulli Experiment Games
Roulette Bernoulli Experiment Simulation
Roulette Bernoulli Experiments
Roulette Bernoulli Experiment Definition

The Canonical Roulette

According to Bernoulli's principle, this faster moving air on the top has a lower pressure than the non-moving air on the bottom. With a greater pressure on the bottom of the paper there is also a. Bernoulli Experiment. Let X denote the result of one binomial trial, where X =1 if you observe a success and X = 0 if you observe a failure. Find the mean and variance of X. Suppose we roll a die until we observe a 6. This is a special case of a negative binomial experiment where r = 1 and p = 1/6. Cool Bernoulli's Principle Science Experiment- BiteSized ExperimentsPresley teaches a simple experiment that shows Bernoulli's Principle in action.

David Little
Mathematics Department Penn State University Eberly College of Science University Park, PA 16802	Office: 403 McAllister Phone: (814) 865-3329 Fax: (814) 865-3735 e-mail:dlittle@psu.edu

In the theory of probability and statistics, a Bernoulli trial is a random experiment with exactly two possible outcomes, “success” and “failure”, in which the probability of success is the same.
Both Bernoulli's equation and the continuity equation are essential analytical tools required for the analysis of most problems in the subject of mechanics of fluids. Purpose: To verify Bernoulli's equation by demonstrating the relationship between pressure head and kinetic head. Bernoulli's apparatus (Figure 1).

Canonical Roulette

Spin - spins the wheel; once the wheel stops spinning, a ball is released from the left margin of the Distribution Function graph according to the value of the Canonical Roulette. The ball travels horizontally until hitting the graph of the Distribution Function, at which point it falls down vertically onto the histogram.
Random - equivalent to 'Spin', however you don't have to wait for the Canonical Roulette to stop spinning.

Controls

Start/Stop - begin/end collecting data at a very high rate.
Clear Data - removes all data currently displayed in the histogram.
Number of Bins - number of bins used to draw a histogram of the collected data.

Distribution Function
Choose from among several Discrete/Continuous Distributions (see below). Once a distribution has been selected, the corresponding graph will be displayed. This graph is used to create data in the following manner. First, the Canonical Roulette is used to create a random real number between 0 and 1 with uniform distribution. Second, a ball is released along the left boundary of the graph of the Distribution Function. The height of the ball (as a proportional of the height of the left boundary) is given by the outcome of the Canonical Roulette. The ball then travels horizontally to the right until it comes in contact with the graph of the Distribution Function, at which point it falls straight down onto the histogram.

Probability Function
Choose from among several Discrete/Continuous Probability Functions (see below). Once a probability function has been selected, the corresponding graph will be displayed. The graph can be used to compare the histogram of the actual data collected with the theoretical values predicted by the probability function.

Discrete Distributions

Binomial - the number of successes out of n independent trials of a Bernoulli experiment, where p is the probability of success on each trial.
Geometric - the number of trials of a Bernoulli experiment before the first success, where p is the probability of success on each trial.
Hypergeometric - the number of red balls drawn from an urn containing r red balls and b blue balls. The balls are drawn from the urn n at a time without replacement.
Negative Binomial - the number of trials (in excess of r) of a Bernoulli experiment until the rth success, where p is the probability of success on each trial.
Poisson - the number of occurrences of an event over a fixed time interval, where lambda is the average rate at which the event occurs.

Continuous Distributions

Arcsine -
Beta -
Cauchy -
Chi squared -
Exponential -
Gamma -
Log Normal -
Maxwell -
Normal -
Pareto - used to model random variables that take on small values with high probability and large values with low probability. For example, the idea that 80% of the wealth is owned by merely 20% of a population implies that the vast majority of the population has a small personal wealth.
Rayleigh -
Triangular -
Uniform -

Graphs of probability P of not observing independent events each of probability p after n Bernoulli trials vs np for various p. Three examples are shown:
Blue curve: Throwing a 6-sided die 6 times gives 33.5% chance that 6 (or any other given number) never turns up; it can be observed that as n increases, the probability of a 1/n-chance event never appearing after n tries rapidly converges to 0.
Grey curve: To get 50-50 chance of throwing a Yahtzee (5 cubic dice all showing the same number) requires 0.69 × 1296 ~ 898 throws.
Green curve: Drawing a card from a deck of playing cards without jokers 100 (1.92 × 52) times with replacement gives 85.7% chance of drawing the ace of spades at least once.

In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, 'success' and 'failure', in which the probability of success is the same every time the experiment is conducted.^[1] It is named after Jacob Bernoulli, a 17th-century Swiss mathematician, who analyzed them in his Ars Conjectandi (1713).^[2]

Roulette Bernoulli Experiment Games

The mathematical formalisation of the Bernoulli trial is known as the Bernoulli process. This article offers an elementary introduction to the concept, whereas the article on the Bernoulli process offers a more advanced treatment.

Since a Bernoulli trial has only two possible outcomes, it can be framed as some 'yes or no' question. For example:

Is the top card of a shuffled deck an ace?
Was the newborn child a girl? (See human sex ratio.)

Therefore, success and failure are merely labels for the two outcomes, and should not be construed literally. The term 'success' in this sense consists in the result meeting specified conditions, not in any moral judgement. More generally, given any probability space, for any event (set of outcomes), one can define a Bernoulli trial, corresponding to whether the event occurred or not (event or complementary event). Examples of Bernoulli trials include:

Flipping a coin. In this context, obverse ('heads') conventionally denotes success and reverse ('tails') denotes failure. A fair coin has the probability of success 0.5 by definition. In this case there are exactly two possible outcomes.
Rolling a die, where a six is 'success' and everything else a 'failure'. In this case there are six possible outcomes, and the event is a six; the complementary event 'not a six' corresponds to the other five possible outcomes.
In conducting a political opinion poll, choosing a voter at random to ascertain whether that voter will vote 'yes' in an upcoming referendum.

Definition[edit]

Independent repeated trials of an experiment with exactly two possible outcomes are called Bernoulli trials. Call one of the outcomes 'success' and the other outcome 'failure'. Let ${displaystyle p}$ be the probability of success in a Bernoulli trial, and ${displaystyle q}$ be the probability of failure. Then the probability of success and the probability of failure sum to one, since these are complementary events: 'success' and 'failure' are mutually exclusive and exhaustive. Thus one has the following relations:

{displaystyle p=1-q,quad quad q=1-p,quad quad p+q=1.}

Roulette Bernoulli Experiment Simulation

Alternatively, these can be stated in terms of odds: given probability p of success and q of failure, the odds for are ${displaystyle p:q}$ and the odds against are ${displaystyle q:p.}$ These can also be expressed as numbers, by dividing, yielding the odds for, ${displaystyle o_{f}}$ , and the odds against, ${displaystyle o_{a}:}$ ,

{displaystyle {begin{aligned}o_{f}&=p/q=p/(1-p)=(1-q)/qo_{a}&=q/p=(1-p)/p=q/(1-q)end{aligned}}}

These are multiplicative inverses, so they multiply to 1, with the following relations:

{displaystyle o_{f}=1/o_{a},quad o_{a}=1/o_{f},quad o_{f}cdot o_{a}=1.}

In the case that a Bernoulli trial is representing an event from finitely many equally likely outcomes, where S of the outcomes are success and F of the outcomes are failure, the odds for are ${displaystyle S:F}$ and the odds against are ${displaystyle F:S.}$ This yields the following formulas for probability and odds:

{displaystyle {begin{aligned}p&=S/(S+F)q&=F/(S+F)o_{f}&=S/Fo_{a}&=F/Send{aligned}}}

Note that here the odds are computed by dividing the number of outcomes, not the probabilities, but the proportion is the same, since these ratios only differ by multiplying both terms by the same constant factor.

Random variables describing Bernoulli trials are often encoded using the convention that 1 = 'success', 0 = 'failure'.

Closely related to a Bernoulli trial is a binomial experiment, which consists of a fixed number ${displaystyle n}$ of statistically independent Bernoulli trials, each with a probability of success ${displaystyle p}$ , and counts the number of successes. A random variable corresponding to a binomial is denoted by ${displaystyle B(n,p)}$ , and is said to have a binomial distribution.The probability of exactly ${displaystyle k}$ successes in the experiment ${displaystyle B(n,p)}$ is given by:

{displaystyle P(k)={n choose k}p^{k}q^{n-k}}

where ${displaystyle {n choose k}}$ is a binomial coefficient.

Bernoulli trials may also lead to negative binomial distributions (which count the number of successes in a series of repeated Bernoulli trials until a specified number of failures are seen), as well as various other distributions.

Roulette Bernoulli Experiments

When multiple Bernoulli trials are performed, each with its own probability of success, these are sometimes referred to as Poisson trials.^[3]

Example: tossing coins[edit]

Consider the simple experiment where a fair coin is tossed four times. Find the probability that exactly two of the tosses result in heads.

Solution[edit]

For this experiment, let a heads be defined as a success and a tails as a failure. Because the coin is assumed to be fair, the probability of success is ${displaystyle p={tfrac {1}{2}}}$ . Thus the probability of failure, ${displaystyle q}$ , is given by

{displaystyle q=1-p=1-{tfrac {1}{2}}={tfrac {1}{2}}}

Using the equation above, the probability of exactly two tosses out of four total tosses resulting in a heads is given by:

{displaystyle {begin{aligned}P(2)&={4 choose 2}p^{2}q^{4-2}&=6times left({tfrac {1}{2}}right)^{2}times left({tfrac {1}{2}}right)^{2}&={dfrac {3}{8}}.end{aligned}}}

References[edit]

Roulette Bernoulli Experiment Definition

^Papoulis, A. (1984). 'Bernoulli Trials'. Probability, Random Variables, and Stochastic Processes (2nd ed.). New York: McGraw-Hill. pp. 57–63.
^James Victor Uspensky: Introduction to Mathematical Probability, McGraw-Hill, New York 1937, page 45
^Rajeev Motwani and P. Raghavan. Randomized Algorithms. Cambridge University Press, New York (NY), 1995, p.67-68

External links[edit]

Wikimedia Commons has media related to Bernoulli trial.

'Bernoulli trials', Encyclopedia of Mathematics, EMS Press, 2001 [1994]
'Simulation of n Bernoulli trials'. math.uah.edu. Retrieved 2014-01-21.

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