Biology Reference
In-Depth Information
I. PROBABILITY BACKGROUND
Consider an experiment whose outcome depends on chance—such as
flipping a coin or rolling a die, the number of chickens hatching on a
farm each day, or how long after sunset a bat leaves a cave. If we repeat
such experiments, the outcomes will vary randomly, and so we say the
outcome is described by a random variable. Each time the experiment is
performed, the random variable takes a specific value corresponding to
that outcome. In most cases, this assignment is very natural. When
rolling a die, the value rolled (1, 2, 3, 4, 5, or 6) will be the number
assigned to the outcome. For some experiments, specific values may be
more likely to occur than others. The chances of an expectant mother
delivering twins are smaller than for delivering a single baby, but are
higher than for septuplets. Each random variable has a probability
distribution specifying how likely an outcome, or a set of outcomes, is to
occur. The probability distribution of the random variable can be
visualized as follows: If we make many (theoretically, infinitely many)
trials and find the numbers the random variable assigns to the trials, the
relative frequency histogram of these numbers will provide a good
approximation of the random variable's distribution. It is common in
probability to denote random variables by letters from the Greek
alphabet, such as
x
and
.
In the previous chapter, we described how Nilsson-Ehle used this
approach to obtain the distribution of phenotypes in the F
2
generation
originating from a parental cross of white and red grain wheat and
presented the results in a table (reproduced here as Table 4-1). The last
column of Table 4-1 represents the probability distribution obtained
from the approximate proportions 1:4:6:4:1 he observed between the
phenotypes in the F
2
. In this example, the random variable representing
the grain color in the F
2
is discrete, as there is a finite number of values
this random variable can take. The same will be true for the random
variable giving the number of chickens hatching on a farm each day. The
Phenotype
(Grain Color)
Number of
Sequences
Genotypes
Probability
WWWW
White
1
1/16
RWWW, WRWW, WWRW,
WWWR
Light pink
4
1/4
RRWW, RWRW, RWWR,
WRRW,WRWR, WWRR
Pink
6
3/8
RRRW, RRWR, RWRR,
WRRR
Dark pink
4
1/4
RRRR
Red
1
1/16
TABLE 4-1.
Genotypes and phenotypes predicted by the polygenic hypothesis.
