Flipping a coin

The term Monte Carlo is often used to describe a wide variety of numerical techniques that are applied to solve mathematical problems by means of the simulation of random variables. The intuitive concept of a random variable is a simple one It is a variable that may take a given value of a set, but we do not know in advance which value it will take in a concrete case. The simplest example at hand is that of flipping a coin. We know that we will get head or tail, but we do not know which of these two cases will result in the next toss. Experience shows that if the coin is a fair one and we flip it many times, we obtain an average of approximately half heads and half tails. So we say that the probability p to obtain a given side of the coin is k A random variable is defined in terms of the values it may take and the related probabilities. In the example we consider, we may write... 

How does phosphorylation affect the activity of phosphofructo-2-kinase (PFK-2), the enzyme that synthesizes fructose 2,6-bisphosphate, a regulator of glycolysis There are two possible answers it either activates it or inactivates it. The simplest approach to the question is just to flip a coin. You should stand a 50 50 chance of getting it right. The next simplest way is to figure it out. 

If flow occurred in an open channel with no particles or fibers and if there were no other mixing mechanisms, all particles would transit the same distance from beginning to end. In a packed bed, each time a molecule or atom encounters a particle or fiber it must go around it to continue on. It is analogous to encounter a tree in a field—one either walks around it to the right or the left and that is equivalent to flipping a coin. Some molecules will encounter more particles than others as illustrated in the following scheme where each encounter causes a chance in direction and the path of a hypothetical molecule is traced by a line (Scheme 1). 

The Problem When you roll a die and flip a coin at the same time, what is the probability that you get an even number on the die with tails ... 

One wants to do better than just flip a coin. 

The choice of the assignment of products to cells (i.e., whether the prechange lot or the postchange lot is assigned to the upper left comer cell of the apparatus) may either be made systematically (i.e., alternate the pattern for each successive run) or randomly (i.e., flip a coin or use some other random mechanism). 

When you flip a coin, the probability of its landing on each side is p = q = I in Equations 28-2 and 28-3. If you flip it n times, the expected number of heads equals the expected number of tails = np = nq = n. The expected standard deviation for n flips is crn = /npq. From Table 4-1, we expect that 68.3% of the results will lie within lrr and 95.5% of the results will lie within 2cr . 

Exercise. Flip a coin N times. Prove that the probability that heads turn up exactly n times is... 

One of the most basic problems in statistics is called the random walk problem. Suppose you take a total of N steps along a north-south street, but before each step you flip a coin. If the coin comes up heads, you step north if the coin comes up tails, you step south. What is the probability that you will end up M steps north of your starting point (in other words, the probability that you will get M more heads than tails) ... 

The winner is the candidate with the most electoral votes. There is a slight chance that there will be a tie, in which case you d have to flip a coin. If you want to play again, press RUN/STOP-RESTORE and type RUN. 

Here Iq is the incident radiation and A is the true absorbance (base e) for a molecularly dispersed system. The probability, P(n) is the same as that for flipping a coin Z times amd coming up with n heads. Thus ... 

When the possible outcomes of a probability experiment are continuous (for example, the position of a particle along the x-axis) as opposed to discrete (for example, flipping a coin), the distribution of results is given by the probability density function P(x). The product P(x)dx gives the probability that the result falls in the interval of width dx centered about the value x. The first condition, that the probability density must be normalized, ensures that probability density is properly defined (see Appendix A6), and that all possible outcomes are included. This condition is expressed mathematically as... 

Indifference is a knife-edge property, rarely observed in nontrivial real-life situations. Incommensurability is, I believe, a vastly more important source of indeterminacy. Consider for instance a customer w ho cannot make up his mind whether to buy car brand A or brand B. If he were indifferent, a one dollar discount on A should make him decisively prefer A. If he still cannot make up his mind, as will typically be the case, the two brands must be incommensurate. To choose, he may flip a coin - or consult his "gut feelings."... 

If you flip a coin 10 times, how many heads will you get Try it, and record your results. Repeat the experiment. Are your results the same Ask friends or members of your class to perform the same experiment and tabulate the results. The table below contains the results obtained by several classes of analytical chemistry students over the period from 1980 to 1998. 

Ratios can represent probabilities, also called odds. This is a ratio that compares the number of ways a certain outcome occurs to the number of outcomes. For example, if you flip a coin 100 times, what are the odds that it will come up heads There are two possible outcomes, heads or tails, so the odds of coming up heads are 50 100. Another way to say this is that 50 out of 100 times the coin will come up heads. In its simplest form, the ratio is 1 2. 

A thought question about random fractals) Redo the previous question, except add an element of randomness to the process to generate from, flip a coin if the result is heads, delete the second quarter of every interval in 5 if tails, delete the third quarter. The limiting set is an example of a random fractal. 

The distribution (7.25) is called binomial. Its most frequent textbook example is the outcome of flipping a coin with probabilities to win and lose given by Pr and pi, respectively. The probability to have nr successes out of A coin flips is then given by the binomial distribution (7.25). 

Apraxia loss of ability to leam or to carry out specific movements, eg, unable to flip a coin when asked to do so... 

Kirkwood s description is not wrong, but it is one-sided. His memorable simplicity conceals a can of worms. The molecular action of vitamin C is as simple and repetitive as flipping a coin, yet the effects are varied, unpredictable and utterly dependent on the milieu in which it operates. Just as flipping a coin leads to diametrically opposed outcomes, so too, vitamin C may on the one hand protect against illness and on the other kill tumours, or even people. The food chemist William Porter summed up the conundrum nicely, rising to an anguished eloquence rarely matched in scientific journals "Of all the paradoxical compounds, vitamin C probably tops the list. It is truly a two-headed Janus, a Dr Jekyll-Mr Hyde, an oxymoron of antioxidants."... 

Such a diverse array of functions lends vitamin C an aura of magic. Yet in each of these cases, vitamin C behaves in exactly the same way at the molecular level, as repetitively as flipping a coin, even if the outcomes are opposed. To see how, let us take a single example in a little more detail. Collagen synthesis illustrates not only how vitamin C works, but also throws light on its antioxidant properties, as well as its more dangerous side. 

Flipping a coin to decide which movie to see. The other movie is better. (Inconsequential) High. (50%) Acceptable. 

Each of these probabilities p, through p has some variability. Just as with flipping a coin, we would not expect the first few trials to yield the same fraction of specific outcomes as would a large number of trials. So, to model the effects of estradiol on target cells requires looking at the outcomes of a given number of trials and noting the results. 

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