Among
40 participants in a training, 8 were youths (less than 30 years), 25 were
adults (30 to 59 years) and remaining 7 were senior citizens (60 or more years).
2 participants are selected at random one after another without replacement of
the name of the first selected participant. Calculate the probability that one
of two participants is a youth.
Multi-categories of a random
variable can be reduced to two categories of an outcome in dependent experiments
or trails without replacement. This is important because two-category discrete probability distribution is most commonly as one is concerned with the probability of a successful or failure event. Thus, Hypergeometric distribution function can
be used instead of Multivariate Hypergeometric distribution function. I will discuss the process of reducing
the multi-categories to two-categories and calculating the two-category
discrete probability distribution without replacement using tree diagram,
formula and Excel software function. For details on two-category and multi-category
discrete probability distributions refer to my statistical notes from 17 to 25.
Tree
Diagram
Tree
diagram is an important means to visualize, count the number of outcomes and
calculate the probability. There are
only two categories, success and failure in each selection of a participant without
replacement. Let X be an event that a youth is selected in a random selection
of a participant, also referred to as the success. The marginal probability of
X, denoted by P(X) equal to p, is eight youths divided by total of 40 participants,
which is equal to 0.20 in the first selection of a participant. Let Y be an
event failure that will consist of other participants (adults and senior
citizens). Thus, the marginal probability of failure, denote by P(Y) equal to q,
is 32 other aged participants divided by total of 40 participants, equal to 0.80
in the first selection of a participant. The conditional probabilities P(X/X),
P(Y/X), P(X/Y) and P(Y/Y) in the second selection of a participant without
replacement remains different than the marginal probabilities in the first
selection of a participant (Diagram 1).
Diagram 1: Marginal and conditional probabilities in two age grouped participants sampled without replacement
The
joint probability that an event X appears in the first selection and an event Y
appears on the second roll, denoted by P(X∩Y), is the product of P(X) and P(Y/X)
which is equal to (8X32) divided by (40X39), 0.1641. Likewise, the joint
probability that an event Y appears in the first roll and an event X appears on
the second roll, denoted by P(Y∩X), is the product of P(Y) and P(X/Y) which is also
equal to (8X32) divided by (40X39), 0.1641. Thus, the total probability of an outcome
that X of two events occurs, is the sum of P(X∩Y) and P(Y∩X), which is equal to
0.3282.
Formula
Formula
is another means of calculating the discrete probability distribution. Let X be
a random variable of interest that takes one of 0, 1 or 2 values as the number
of youths in the sample of two participants sampled without replacement,
denoted by ‘x’. The probability distribution of X depends on the parameters,
‘n’, ‘M’ and ‘N’, and is given by the expression
P(X=x) = h(x;n,M,N) = [C(M,x) X C(N-M,n-x)]/C(N,n)
This
distribution is referred to as Hypergeometric distribution.
In
this example, n=2, M=8 and N=40 and ‘x’ takes the value 1. Putting these values
in the above formula, one gets
P(X=1)
= [C(8,1) X C(32,1)/C(40,2)] = (8 X 32 X 2)/ (40 X 39) = 0.3282.
EXCEL
Function
Excel
software is commonly available in the desktop or the laptop and is an important
means to calculate the discrete probability distribution. Excel software has
the ‘HYPGEOM.DIST’ function that has five fields ‘Sample-s’, ‘Number_sample’,
‘Population_s’, ‘Number_pop’ and Cumulative’ as shown in
the function argument box of Diagram 2.
Diagram 2: ‘HYPERGEOM.DIST’ Function Arguments Using Dataset in Excel Worksheet and using ‘FALSE’ logical value in the field ‘Cumulative’
The
field ‘Sample_s’ takes the number of successes in trials. In this
example, this field takes the value one youth as shown in the cell B3 in the table
as well as an argument box in Diagram 2. The field ‘Number_sample’ is two
participants sampled without replacement. The field ‘Population_s’ is
the number of successes in the population. This example has 8 youths out of 40
participants. The field ‘Number_pop’ is the population size, the total
of 40 participants. The field ‘Cumulative’ is a logical value that
determines the form of the function. If ‘Cumulative’ is ‘FALSE’, ‘HYPGEOM.DIST’
calculates the probability mass function (PMF), which gives the probability associated
with the value assigned to the field ‘Sample_s’ as the number of successes.
Fixing all five fields
in the function arguments, ‘HYPGEOM.DIST’ function calculated the PMF equal to
0.3282. It means that there is 32.8 percent
chance that one of two participants sampled without replacement will be a youth.
The probability calculated using Excel software function is
equal to the values calculated in tree diagram and formula sections above. Discussion in this note
indicates that the multi-category can be reduced to two-category in which one
category will be considered as a successful event and another as a failure
event. Then, Hypergeometric distribution can be applied to two category discrete
probability distribution without replacement. This will limit the use of Multivariate
Hypergeometric Probability Distribution function. Another learning is that both manual and auto
calculation produce the same values and are useful to calculate the discrete
probability distribution without replacement. Conceptual understanding is a
backbone and automatization is efficient. Thus, both are important knowledge
and skill sets.
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