LAWS OF PROBABILITY

• Jurisdiction: Australia

LAWS OF PROBABILITY

First law of probability ................................................................................... [80A.1000]

Second law of probability.............................................................................. [80A.1020]

Joint probabilities .......................................................................................... [80A.1040]

Third law of probability (conditional probabilities)......................................... [80A.1060]

Likelihood Ratios ........................................................................................... [80A.1080]

Genotype frequency and match probability .................................................. [80A.1100]

[80A.1000] First law of probability

All probabilities lie between 0 and 1. An event with a zero probability never occurs: a probability of 1 equates to certainty.

[80A.1020] Second law of probability

This leads to a mathematical way of combining probabilities of alternative (mutually exclusive) events. The probability of any one of several mutually exclusive events occurring can be obtained by adding together their individual probabilities. This is a form of the second law of probability (as illustrated above at [80A.800] in the two throws of the dice).

Early versions of a DNA profiling kit (the Amplitype™ HLA DQa Forensic DNA Amplification and typing kit) marketed by Perkin Elmer enabled six alleles (called types 1.1, 1.2, 1.3, 2, 3 and 4) of the gene HLA DQa to be differentiated. A subsequent kit (Polymarker) enabled type 4 alleles to be classified as either type 4.1 or a combined class comprising types 4.2 and 4.3 (which could not be differentiated from each other using either kit).

Laboratories that used the earlier kit established databases from which the relative frequency of type 4 alleles could be determined. Using the Polymarker kit, the relative frequencies of type 4.1 and (4.2/4.3) could be determined. If no other subtypes of the type 4 allele were observed, the probability of a randomly selected allele being type 4 using the older kit could be estimated from the new database: a type 4 allele would be typed as either type 4.1 or (4.2/ 4.3) using the Polymarker kit. We can write this mathematically:

p(4) = p(4.1) + [p(4.2/4.3)]

where p( ) denotes the probability of the event in brackets.

In statistics, "OR" means "+"

Thus, if there are only two outcomes possible, then the sum of their probabilities is 1. For example, when a coin is tossed, it must come down either heads OR tails (if we assume that the toss would be invalid if the coin landed on its edge). The probability of it landing heads up plus the probability of tails up equals 1.

p(H) + p(T)=1

In general, the sum of the probabilities of all possible outcomes is 1, if the events are mutually exclusive and exhaustive, in that there are no other possible outcomes. Thus, the probability of one of the possible outcomes not occurring is 1 minus the probability that it does occur:

p(H) = 1 - p(T)

where p(T) is the event that the outcome of the toss is not heads.

[80A.1040] Joint probabilities

Suppose we observe two events, A and B, and we want to estimate the probability of the two events occurring simultaneously. There are several ways in which the combination of events could occur (eg, A might precede or follow B; A might occur more often if B occurs). The probability of both A and B occurring, if they are statistically independent events, can be written as

p(A,B) = p(A) × p(B)

In this equation, (A,B) denotes the combined event (both A and B together).

Multiplying the probabilities of the individual events is an example of the "Product Rule": see above [80A.600]. If A and B are alleles that have been observed at a locus, then the probability of a randomly selected person inheriting a paternal allele A and a maternal allele B can be estimated by multiplying their relative frequencies:

p(A,B) = fA × fB

In practice, however, the genotypes of the hypothetical random person's parents are not known, and therefore the genotype A,B could have arisen in either of two ways: the A could have come from the father and the B from the mother, or the B could have come from the father and the A from the mother. If each of these combinations has a probability of fA x fB of occurring, the total probability of observing the genotype A,B is given by:

p(A,B) =2 × fA × fB

[80A.1060] Third law of probability (conditional probabilities)

More generally, the joint probability of two or more events can be described using conditional probabilities. When two or more events occur, they may or may not affect each other.

If the probability of one event, A, depends on the outcome of another event, B, then event A is said to be conditional on B. The conditional probability of A is the probability that A will occur, given that B has already occurred. This conditional probability is written as:

p(A/B)

In this formula, the vertical line ( / ) is read as "given that."

For example, suppose there is a school having 60% boys and 40% girls as students. Eighty percent of the students wear trousers. The boys all wear trousers; the female students wear either trousers or skirts. We want to know the proportion of girls who wear trousers.

The event G is that the student is a girl, and the event T is that the student is wearing trousers. The desired proportion can be rephrased as the probability of a student wearing trousers, given that the student is a girl, and in mathematical terms it is written as:

P(T/G)

The probability of trousers is conditional on the sex of the student: for a girl it is less than certain she is wearing trousers.

It is convenient to list all the possible outcomes of these two events: girls in trousers; girls in skirts; boys in trousers. A tree diagram helps to determine the probability of each outcome:

Tree diagram showing the...