Binomial Probability Example

A safety inspector observes a safety valve 5 times and finds that it is open 4 of the 5 times. Past observations have indicated that there is a 50% chance at any given time that the valve will be open. What is the probability the inspector s observation That is, what is the probability of observing this valve 5 times and finding it open 4 times In the binomial probability formula, n is equal to the number of observations—in this case, the number of times the valve is observed so n = 5. The number of desired 

P is the probability of the binomial event N = 5 (total number of observations) r = 4 (number of times the valve is open) 

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Example 2.17 Binomial Probability fiom Tooth Cavities. Suppose that the incidence of tooth cavities in a population of childien is known to be 10%. A toothpaste manufacturer... 

This is reasonably close to the binomial probability of 0.392 found in Example 2.17. The R code for this example is as follows ... 

The multinomial probability distribution is a generalization of the binomial probability distribution. A binomial distribution describes two-outcome events such as coin flips. A multinomial probability distribution applies to t-outcome events where n, is the number of times that outcome i = 1, 2,3,..., f appears. For example, t = 6 for die rolls. For the multinomial distribution, the number of distinguishable outcomes is given by Equation (1.18) W = AH/lniln in j nj ). The multinomial probability distribution is... 

The binomial probability distribution is an important example of a discrete distribution. Say that we toss an unbiased coin n times and want to find the probability that heads will come up m times. For a single toss, the probability of heads is equal to 1 /2. The probability that heads will come up every time on m consecutive throws is... 

A rise in volatility generates a range of possible future paths around the expected path. The actual expected path that corresponds to a zero-coupon bond price incorporating zero OAS is a function of the dispersion of the rai e of alternative paths around it. This dispersion is the result of the dynamics of the interest-rate process, so this process must be specified for the current term structure. We can illustrate this with a simple binomial model example. Consider again the spot rate structure in Table 12.1. Assume that there are only two possible future interest rate scenarios, outcome 1 and outcome 2, both of equal probability. The dynamics of the short-term interest rate are described by a constant drift rate a, together with a volatility rate a. These two parameters describe the evolution of the short-term interest rate. If outcome 1 occurs, the one-period interest rate one period from now will be... 

The binomial distribution describes a population whose members have only certain, discrete values. A good example of a population obeying the binomial distribution is the sampling of homogeneous materials. As shown in Example 4.10, the binomial distribution can be used to calculate the probability of finding a particular isotope in a molecule. 

Where f(x) is tlie probability of x successes in n performances. One can show that the expected value of the random variable X is np and its variance is npq. As a simple example of tlie binomial distribution, consider tlie probability distribution of tlie number of defectives in a sample of 5 items drawn with replacement from a lot of 1000 items, 50 of which are defective. Associate success with drawing a defective item from tlie lot. Tlien the result of each drawing can be classified success (defective item) or failure (non-defective item). The sample of items is drawn witli replacement (i.e., each item in tlie sample is relumed before tlie next is drawn from tlie lot tlierefore the probability of success remains constant at 0.05. Substituting in Eq. (20.5.2) tlie values n = 5, p = 0.05, and q = 0.95 yields... 

Nature In an experiment in which one samples from a relatively small group of items, each of which is classified in one of two categories, A or >, the hypergeometric distribution can be defined. One example is the probability of drawing two red and two black cards from a deck of cards. The hypergeometric distribution is the analog of the binomial distribution when successive trials are not independent, i.e., when the total group of items is not infinite. This happens when the drawn items are not replaced. 

One defective item in a sample is shown to be most probable but this sample proportion occurs less than four times out of ten, even though it is the same as the population proportion. In the previous example, we would expect about one out of ten sampled items to be defective. We have intuitively taken the population proportion (p = 0.1) to be the expected value of the proportion in the random sample. This proves to be correct. It can be shown that for the binomial distribution ... 

This binomial provides a simple means to consider the initial very low concentrations of ligands and the stochastic aspects of selection. For example, the probability that any ligand i will survive selection can be expressed ... 

Fig. 41.2 Examples of binomial frequency distributions with different probabilities. The distributions show the expected frequency of obtaining n individuals of type A in a sample of 5. Here P is the probability of an individual being type A rather than type B.

Chemical examples of data likely to be distributed in a binomial fashion occur when an observation or a set of trial results produce one of only two possible outcomes for example, to determine the absence or presence of a particular pesticide in a soil sample. To establish whether a set of data is distributed in binomial fashion calculate expected frequencies from probability values obtained from theory or observation, then test against observed frequencies using a x -test or a G-test. 

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). 

Hence, CTS cannot answer the question WiU this trial succeed but rather answers the question What is the probability of this trial succeeding One outcome frequently reported from a CTS is that of statistical power—the probability of declaring the null hypothesis (the hypothesis of no difference) false. Power is computed from a stochastic simulation by counting the number of times the null hypothesis is rejected divided by the total number of replicates used in the simulation. For example, if 88 runs out of 1000 reject the null hypothesis, the power of the trial design is 8.8%. Since this type of analysis is binomial (the trial either fails or it succeeds), the (1 - a)% confidence interval for the power of the trial may be calculated using the normal approximation as... 

The Poisson distribution describes the probability of a discrete number of events occurring within a fixed interval, given that the probability of the event occurring is independent of the size of the interval. For example, suppose that a manufacturing defect occurs with an average rate of occurrence p and the products are manufactured over an interval n. The expected number of defects is clearly np. The Poisson distribution is one way of describing the probability that x defects will occur in an interval n. This distribution is a generalization of the binomial distribution to an infinite number of trials. The mathematical form of the Poisson distribution [2] is... 

The individual terms of this expansion are the probability e m=probability of 0 defectives me m=probability of 1 defective =probability of 2 defectives =probability of 3 defectives By using the same example as for the binomial distribution, we can make the comparison =50... 

Probability distribution. A mathematical model which, for a range of values, associates a probability (or probability density) with a given value. Common examples are the Normal binomial, t, F, chi-square, exponential (or negative exponential) and Poisson distributions. 

In the first step, we determine the interest rate path in which we create a risk-neutral recombining lattice with the evolution of the 6-month interest rate. Therefore, the nodes of the binomial tree are for each 6-month interval, and the probability of an upward and downward movement is equal. The analysis of the interest rate evolution has a great relevance in callable bond pricing. We assume that the interest rate follows the path shown in Figure 11.4. In this example, we assume for simplicity a 2-year interest rate. We suppose that the interest rate starts at time tg and can go up and down following the geometric random walk for each period. The interest rate rg at time tg changes due to two main variables ... 

A simple example involves an investor who is due to receive a single cash flow C at a certain time T, in the future. It is also known that the probability of a default by ABC at any time between today and time T is 5.00%, denoted as pj. There are only two possible outcomes in this scenario, illustrated from the investor s perspective by the following binomial tree ... 

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