A bag contains three balls of different colors (A of Blue
color, B Green and C Red). Select two balls one randomly one at a time and
replace back to the bag after selecting. Calculate the total number of permutation
of possible outcomes containing two balls selected with replacement.
Calculating the total number of outcomes using some objects
selected randomly one at a time with replacement from the total number of
objects is important to calculate the probability of events. Also, important is
a mathematical concept of the ‘Permutation’ used in identifying the total
number of possible outcomes given the condition and calculating the probability
of events. I show the permutation of total number outcomes using the tree
diagram 1.
Diagram 1: Permutation
of Outcomes in Selecting Two Out of Three Balls With Replacement
There will be two stages required to select or arrange two
of three balls. At the first stage or draw, there are three possibilities of randomly
selecting the first ball. The first ball could be one of Blue (A), Green (B) or
Green (C) balls shown by boxes of those letters and colors in Diagram 1. Once
the first ball, Blue (A), is drawn and replaced back, there will be again all
three balls in the bag, A, B and C for the second stage or draw. If the second
ball drawn is again A, the first outcome constituting two balls is AA which is
numbered 1 on the right most side of the diagram 1. Following the same process,
if the second ball drawn is Green (B), the second outcome is AB, numbered 2 in
the diagram 1. In the same way, if the first ball is B and second ball is A,
the outcome is BA which is numbered 4 in the diagram as the outcome AB is
different because the position of the ball, first or second, is meaningful. Thus,
there are nine permutations each with two balls selected randomly with
replacement from three balls. It is noteworthy to mention here that the
digital keys are formulated using the same principle in which only one outcome
or the set of numbers among all possible outcomes will open the digital lock.
Likewise is the case of a password in an email or a website account.
Now, I will let you brainstorm how many will be
the permutation of outcomes with replacement if there are five balls out of
which two balls are drawn one at a time with replacement? Could you use a
diagram to show all possible outcomes? Perhaps, difficult to show. Although,
the diagrammatic logic is important to understand, as the number of balls increases,
the diagram becomes complicated and it will be less possible to show all
outcomes in the diagram. One may need to use the formula. Could you calculate using
a formula? If yes, that is fine. If not, wait for my other statistical notes.
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