Probability

Probability measures the likelihood that something specific will occur. For example, a tossed coin has an equal chance, or probability, of landing with one side up ("heads") or the other ("tails"). If you drive without a seat belt, your probability of being injured in an accident is much higher than if you buckle up. Probability uses numbers to explain chance.

If something is absolutely going to happen, its probability of occurring is 1, or 100 percent. If something absolutely will not happen, its probability of occurring is 0, or 0 percent.

Probability is used as a tool in many areas of genetics. A clinical geneticist uses probability to determine the likelihood that a couple will have a baby with a specific genetic disease. A statistical geneticist uses probability to learn whether a disease is more common in one population than in another. A computational biologist uses probability to learn how a gene causes a disease.

The Clinical Geneticist and the Punnett Square

A Punnett square uses probability to explain what sorts of children two parents might have. Suppose a couple knows that cystic fibrosis, a debilitating respiratory disease, tends to run in the man's family. The couple would like to know how likely it is that they would pass on the disease to their children.

A clinical geneticist can use a Punnett square to help answer the couple's question. The clinical geneticist might start by explaining how the disease is inherited: Because cystic fibrosis is a recessive disease caused by a single gene, only children who inherit the disease-causing form of the cystic fibrosis gene from both parents display symptoms. On the other hand, because the cystic fibrosis gene is a recessive gene, a child who inherits only one copy of a defective gene, along with one normal version, will not have the disease.

Suppose the recessive, disease-causing form of the gene is referred to as "f" and the normal form of the gene is referred to as "F." Only individuals with two disease-causing genes, ff, would have the disease. Individuals with either two normal copies of the gene (FF) or one normal copy and one mutated copy (Ff) would be healthy.

If the clinical geneticist tests the parents and finds that each carries one copy of the cystic fibrosis gene, f, and one copy of the normal gene, F, what would be the probability that a baby of theirs would be born with the cystic fibrosis disease? To answer this question, we can use the Punnett square shown in the figure above. A Punnett square assumes that there is an equal probability that the parent will pass on either of its two gene forms ("alleles") to each child.

The parents' genes are represented along the edges of the square. A child inherits one gene from its mother and one from its father. The combinations of genes that the child of two Ff parents could inherit are represented by the boxes inside the square.

Of the four combinations possible, three involve the child's inheriting at least one copy of the dominant, healthy gene. In three of the four combinations, therefore, the child would not have cystic fibrosis. In only one of the four combinations would the child inherit the recessive allele from both parents. In that case, the child would have the disease. Based on the Punnett square, the counselor can tell the parents that there is a 25 percent probability, or a one-in-four chance, that their baby will have cystic fibrosis.

The Statistical Geneticist and the Chi-Square Test

Researchers often want to know whether one particular gene occurs in a population more or less frequently than another. This may help them determine, for example, whether the gene in question causes a particular disease. For a dominant gene, such as the one that causes Huntington's disease, the frequency of the disease can be used to determine the frequency of the gene, since everyone who has the gene will eventually develop the disease. However, it would be practically impossible to find every case of Huntington's disease, because it would require knowing the medical condition of every person in a population. Instead, genetic researchers sample a small subset of the population that they believe is representative of the whole. (The same technique is used in political polling.)

Whenever a sample is used, the possibility exists that it is unrepresentative, generating misleading data. Statisticians have a variety of methods to minimize sampling error, including sampling at random and using large samples. But sampling errors cannot be eliminated entirely, so data from the sample must be reported not just as a single number but with a range that conveys the precision and possible error of the data. Instead of saying the prevalence of Huntington's disease in a population is 10 per 100,000 people, a researcher would say the prevalence is 7.8-12.1 per 100,000 people.

The potential for errors in sampling also means that statistical tests must be conducted to determine if two numbers are close enough to be considered the same. When we take two samples, even if they are both from exactly the same population, there will always be slight differences in the samples that will make the results differ.

A researcher might want to determine if the prevalence of Huntington's disease is the same in the United States as it is in Japan, for example. The population samples might indicate that the prevalences, ignoring ranges, are 10 per 100,000 in the United States and 11 per 100,000 in Japan. Are these numbers close enough to be considered the same? This is where the Chi-square test is useful.

First we state the "null hypothesis," which is that the two prevalences are the same and that the difference in the numbers is due to sampling error alone. Then we use the Chi-square test, which is a mathematical formula, to test the hypothesis.

The test generates a measure of probability, called a p value, that can range from 0 percent to 100 percent. If the p value is close to 100 percent, the difference in the two numbers is almost certainly due to sampling error alone. The lower the p value, the less likely the difference is due solely to chance.

Scientists have agreed to use a cutoff value of 5 percent for most purposes. If the p value is less than 5 percent, the two numbers are said to be significantly different, the null hypothesis is rejected, and some other cause for the difference must be sought besides sampling error. There are many statistical tests and measures of significance in addition to the Chi-square test. Each is adapted for special circumstances.

Another application of the Chi-square test in genetics is to test whether a particular genotype is more or less common in a population than would be expected. The expected frequencies can be calculated from population data and the Hardy-Weinberg Equilibrium formula. These expected frequencies can then be compared to observed frequencies, and a p value can be calculated. A significant difference between observed and expected frequencies would indicate that some factor, such as natural selection or migration, is at work in the population, acting on allele frequencies. Population geneticists use this information to plan further studies to find these factors.

The Computational Biologist and BLAST

Genetic counseling lets potential parents make an informed decision before they decide to have a child. Geneticists, however, would like to be able to take this one step further: They would like to be able to cure genetic diseases. To be able to do so, scientists must first understand how a disease-causing gene results in illness. Computational biologists created a computer program called BLAST to help with this task.

To use BLAST, a researcher must know the DNA sequence of the disease-causing gene or the protein sequence that the gene encodes. BLAST compares DNA or protein sequences. The program can be used to search many previously studied sequences to see if there are any that are similar to a newly found sequence. BLAST measures the strength of a match between two sequences with a p value. The smaller the p value, the lower the probability that the similarity is due to chance alone.

If two sequences are alike, their functions may also be alike. For BLAST to be most useful to a researcher, there would be a gene that has already been entered in the library that resembles the disease-causing gene, and some information would be known about the function of the previously entered gene. This would help the researcher begin to hypothesize how the disease-causing gene results in illness.

Rebecca S. Pearlman

Bibliography

Nussbaum, Robert L., Roderick R. McInnes, and Huntington F. Willard. Thompson & Thompson Genetics in Medicine, 6th ed. St. Louis, MO: W. B. Saunders, 2001.

Purves, William K., et al. Life: The Science of Biology, 6th ed. Sunderland, MA: Sinauer Associates, 2001.

Seidman, Lisa, and Cynthia Moore. Basic Laboratory Methods for Biotechnology: Textbook and Laboratory Reference. Upper Saddle River, NJ: Prentice-Hall, 2000.

Tamarin, Robert H. Principles of Genetics, 7th ed. Dubuque, IA: William C. Brown,2001.

Internet Resources

The Dolan DNA Learning Center. Cold Spring Harbor Laboratory. <http://vector.cshl.org>.

The National Center for Biotechnology Information. <http://www.ncbi.nlm.nih.gov>.