Coin tossing

Statistical considerations make it possible to test the assumption of independent additions. Let us approach this topic by considering an easier problem coin tossing. Under conditions where two events are purely random-as in tossing a fair coin-the probability of a specific sequence of outcomes is given by the product of the probabilities of the individual events. The probability of tossing a head followed by a head-indicated HH-is given by... 

Parameters As a way of characterizing probabihty functions and densities, certain types of quantities called parameters can be defined. For example, the center or gravity of the distribution is defined to be the population mean, which is designated as [L. For the coin toss [L =. 5, which corresponds to the average value of x i.e., for half of the time X will take on a value 0 and for the other half a value 1. The average would be. 5. For the spinning wheel, the average value would also be. 5. 

Deterministic Randomness. On the one hand, equation 4.8 is a trivial linear difference equation possessing an equally as trivial solution for each initial point Xq Xn = 2"a o (mod 1). Once an initial point is chosen, the future iterates are determined uniquely. As such, this simple system is an intrinsically deterministic one. On the other hand, look again at the binary decimal expansion of a randomly selected a o- This expansion can also be thought of as a particular semi-infinite sequence of coin tosses. 

The assay sashay is like betting on coin tosses with the following rules. We toss two coins. If the first one comes up heads, I win, and the second coin is irrelevant. If the first one comes up tails, we have to decide whether the toss counts. To do that, we look at the second coin. If it also comes up tails, the toss does not count and we call it a draw. With these rules, I will win 50 per cent of the time and it will be a draw 25 per cent of the time. You win only if both the first coin comes up heads and the second comes up tails, which will only happen 25 per cent of the time. So the odds are heavily stacked in my favour. If you doubt this, please get in touch. I will be happy to play you for real money. Using these standards to judge the effectiveness of a medication is voodoo science to the nth degree. 

A full cycle of walk and coin toss is then described by a unitary matrix... 

Normal males inherit an X chromosome from their mother and a Y chromosome from then-father, whereas normal females inherit an X chromosome from each parent. Because the Y chromosome carries only about 30 protein-coding genes and the X chromosome carries hundreds of protein-coding genes, a mechanism must exist to equalize the amount of protein encoded by X chromosomes in males and females. This mechanism, termed X inactivation, ocairs very early in the development of female embryos. When an X chromosome is inactivated, its DNA is not transcribed into mRNA, and it is visuahzed under the microscope as a highly condensed Barr body in the nuclei of interphase cells. X inactivation has several important characteristics It is random—in some cells of the female embryo, the X chromosome inherited from the father is inactivated, and in others the X chromosome inherited from the mother is inactivated. Like coin tossing, this is a random process. 

Alleles of lod that are dose together on the same chromosome are likely to be inherited together these lod are said to be linked. If two lod are on different chromosomes, or if they are far apart on the same chromosome, their aUdes will be transmitted independently. This means that if an allele at one locus is transmitted, there is a 50% chance (as in coin tossing) that a given allde at the other locus will also be transmitted. Linked loci are dose enough together so that the chance of a recombination is less than 50%. Thus, their inheritance is not independent. 

The treatment each trial participant receives is often decided by a process called randomization. This process can be compared to a coin toss that is done by computer. During clinical trials, no one likely knows which therapy is better, and randomization assures that treatment selection will be free of any preference a physician may have. Randomization increases the likelihood that the groups of people receiving the test drug or control are comparable at the start of the trial, enabling comparisons in health status between groups of patients who participated in the trial. 

The tossing of a coin is an experiment whose outcome is a random variable. Intuitively we assume that all coin tosses occur from an underlying population where the probability of heads is exactly 0.5. However, if we toss a coin 100 times, we may get 54 heads and 46 tails. We can never verify our intuitive estimate exactly, although with a large sample we may come very dose. 

We can calculate expectation values for this continuous distribution in much the same way as we calculated them in the last section for ten coin tosses. The generalization of Equation 4.8 for a continuous distribution is ... 

In general, as we increase the number of coin tosses, the absolute expected deviation from exactly N/2 heads I (M2) = j grows, but the fractional expected... 

These results turn out to be applicable to much more than just coin toss problems. A few examples are given below. 

Under these assumptions, the probability of finding a carbon dioxide molecule MX from its starting point after N collisions is mathematically exactly the same as the coin toss probability of M more heads than tails. The root-mean squared distance traveled... 

We showed in the last section that there is a 95% chance that after 10,000 coin tosses, the number of heads would be between 4902 and 5098. The most common definition treats error bars as such a 95% confidence limit. Table 4.1 shows that 95% of the area in a Gaussian is within 1.96error bars are 1.96a and the number of heads hr after 10,000 coin tosses is 5000 98. For the reasons discussed below, we would probably round it to 5000 100. So we would not be sufficiently surprised by 5050 heads to judge that something was wrong we would be surprised enough by 5200 heads. 

Sometimes a measurement involves a single piece of calibrated equipment with a known measurement uncertainty value o, and then confidence limits can be calculated just as with the coin tosses. Usually, however, we do not know o in advance it needs to be determined from the spread in the measurements themselves. For example, suppose we made 1000 measurements of some observable, such as the salt concentration C in a series of bottles labeled 100 mM NaCl. Further, let us assume that the deviations are all due to random errors in the preparation process. The distribution of all of the measurements (a histogram) would then look much like a Gaussian, centered around the ideal value. Figure 4.2 shows a realistic simulated data set. Note that with this many data points, the near-Gaussian nature of the distribution is apparent to the eye. 

This formula is easy to verify with the coin toss distribution. As discussed above, with 10,000 coin tosses you have 5000 98 heads. You can also verify (by calculating a) that with 2500 coin tosses you have 1250 49 heads, but for 12,500 tosses you get 6250 109.56733 heads, not 6250 147 heads as you would get by just adding the error bars for 10,000 tosses and 2500 tosses. We would report 6250 110 heads, using the rounding off convention discussed above (last digit uncertain between 3 and 30 units). 

The quantity fJ> (Et — Ej) plays the same role in Equation 4.26 as a played in Equation 4.25 (in the coin toss problem, the energy difference between adjacent levels was one coin). Increasing /3 increases the fractional population of highly excited states, and thus increases the total energy of the system. Equation 4.26 is called the Boltzmann distribution, after Ludwig Boltzmann, a famous theoretical physicist. 

M2 was for the coin toss distribution in Section 4.2 this quantity is just equal to o2 for a Gaussian. We thus have... 

Use the binomial distribution to calculate the exact probability of getting 95 or more heads out of 100 coin tosses. 

Show that 95% of the time, // after 10,000 coin tosses will be between 4902 and 5098. 

Find the probability of getting 50.2% heads out of 1,000,000 tosses, and show that it is the same as the probability of getting more than 52% heads after 10,000 coin tosses. 

When pollsters quote error bars or likely errors, they are actually quoting 95% confidence limits. If the percentages in the polls are around 50%, this reduces to a coin toss problem just like the ones discussed in this chapter. Suppose 100 people are asked to compare French and California wines, and the preferences are exactly evenly divided. You could report that 50% prefer French wine. What would be the error bars, given the assumptions above ... 

However, since one electron or proton has two possible spin states, N electrons or protons have 2N possible states (just like the coin toss case). Molecules can be configured to be simultaneously in many different quantum states, just as the electron in the two-slit experiment seems to pass through both slits simultaneously. In principle, this property can be used someday to make massively parallel computers, and such computers with five or six bits have been made in the laboratory (using NMR). As of this writing, nobody knows whether or not it will ever be possible to build a quantum computer which is big enough to do a computation faster than a conventional machine, although it is clear that NMR will not work for this application. 

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