# Experimenting with the probability of things

Make a table with the different options: Look away and choose one marble from each cup. What are the chances that it will be a boy or a girl? If you choose one marble from each container again and again, how many boys and how many girls will you get?

Common Distributions in Biology A few simple distributions provide reasonable models for a variety of biological experiments. Consider an experiment that involves some fixed number of independent trials.

Each trial has two possible outcomes, conveniently termed success and failure, and the probability of success is a fixed value. If you draw 5 members of the population at random, what is the probability that 3 of them will display the dominant phenotype?

We assume that the population is sufficiently large that removing 5 individuals is without effect.

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If the gene frequency is 0. Since the trials are independent we can obtain the probability for this outcome by multiplying the probabilities of the individual events What we really want is the probability of obtaining 3 successes irrespective of order The generalization of the above example is the binomial distribution.

A fixed number, N, of independent trials are conducted, each of which may have one of two possible outcomes e. The formal derivation of the Poisson distribution concerns events that occur with constant density in time or space an example is radioactive decay.

In biological experiments this distribution commonly arises as an approximation to the binomial distribution where the number of trials, N, is very large and the success probability, p, is very small. What is the probability that you will get at least two resistant colonies? In this case, the expected number of colonies, m, is 4.

The discrete uniform distribution applies when all of the events in the sample space occur with equal probability. This simple distribution is useful because we can frequently transform data obtained from complex experiments so that it conforms to this distribution. This ability will be useful for hypothesis testing.

If you are daring and good at algebra, you might try deriving it.

Theoretical vs. Experimental Probability When asked about the probability of a coin landing on heads, you would probably answer that the chance is ½ or . One of the most uncomfortable things that students don’t like about formulas is the lack of them in probability and statistics. There are a few staples, including some must-know notation: Probability Range 0 ≤ P(A) ≤ 1 This states that the probability of an event is somewhere between zero and % (as a decimal, that’s 0 and 1). Things like experiment, sample space and event to name a few. A lot of times people associate probability with gambling, like playing cards and lotto. It can be used to find out your chances of winning:) or losing: (a game of chance.

An interesting variation on the binomial distribution is to consider a sequence of Bernoulli trials in which N is not fixed, but is the number of trials required to obtain r "successes".

You want two mutants to study in detail.

## The Feynman Lectures on Physics Vol. III Ch. 3: Probability Amplitudes

Because the apparatus you use to measure this phenotype can only accommodate a single animal, you test animals sequentially from the litter.

What is the probability that you will obtain at least two mutants in no more than four chosen progeny? Thus, the probability that you will have at least 2 mutants in the first 4 tested animals is 0.

A condition of the binomial distribution section 2. This condition can only be met if the trials are independent, i.

## Let's Toss a Coin!

If instead our sample is taken from a finite population without replacement, the success probability for a particular trial depends on the outcomes of the preceding trials. For k small relative to the population size N, the frequencies approach those given by the binomial distribution.

A binomial experiment is an experiment which satisfies these four conditions A fixed number of trials Each trial is independent of the others There are only two outcomes The probability of each outcome remains constant from trial to trial. These can be summarized as: An experiment with a fixed. Schaum's Outline of Probability and Statistics 36 CHAPTER 2 Random Variables and Probability Distributions (b) The graph of F(x) is shown in Fig. The following things about the above distribution function, which are true in general, should be noted. An event is a collection of possible outcomes of an experiment. An event is said to occur as a result of an experiment if it contains the actual outcome of that experiment. Individual outcomes comprising an event are said to be favorable to that event. Events are assigned a measure of certainty which is called probability (of an event.).

You collect a group of 35 animals, of which 15 are mutant and 20 wild-type, and allow them to age. Of the last 10 animals to expire, 8 are mutant.

What is the probability that 8 or more of the last 10 mice to die will be mutant if deaths occur at random in the population as a whole?

## Properties of Probability Distributions

Thus, the probability that exactly 8 of the mice are mutant is Similarly, f 9 and f 10 are 5.Page 1 of 2 Chapter 12 Probability and Statistics Probability of Independent and Dependent Events PROBABILITIES OF INDEPENDENT EVENTS Two events are if the occurrence of one has no effect on the occurrence of the other.

Conditional probability is denoted by the following: The above is read as the probability that B occurs given that A has already occurred. The above is mathematically defined as: Set Theory in Probability. A sample space is defined as a universal set of all possible outcomes from a given experiment.

Pedagogical use of computer programs: Probability theory makes predictions about experiments whose outcomes depend upon chance. Consequently, it lends itself beautifully to the use of computers as a mathematical tool to simulate and analyze chance experiments.

In the text the computer is utilized in several ways. First, it provides a labora-. The probability of 60 correct guesses out of is about %, which means that if we do a large number of experiments flipping coins, about every 35 experiments we can expect a score of 60 or better, purely due to chance.

This is the first in a series of products which bring probability to life in middle school math classrooms! Students perform six probability experiments, tabulate data in custom-made worksheets, calculate experimental probability, then tabulate all possible outcomes to compute theoretical probability.

Probability Dice Game (58 ratings) With all of the different outcomes that may result from a single roll, dice are the perfect way to introduce probability math.

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