A bag contains four balls
of different colors (A of Red color, B Blue, C Green and D yellow). Select two
balls randomly one at a time and do not replace back to the bag after selecting.
Calculate the total number of permutation of possible outcomes containing two balls
selected without replacement.
Calculating the total number of
outcomes using some objects selected randomly one at a time without replacement
from the total number of objects is important to calculate the probability of
events. Because, in sampling some sampling units are selected without
replacement from the population. 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 favorable outcomes using the tree diagram 1.
Diagram 1: Permutation of Outcomes in Selecting Two
Out of Four Balls Without Replacement
There will be two stages required
to select or arrange two out of four balls. At the first stage or draw, there
are four possibilities of randomly selecting the first ball. The first ball
could be one of red (A), blue (B), green (C) or yellow (D) balls shown by boxes
of those letters and colors in Diagram 1. Once the first ball, red (A), is
drawn and not replaced back, there will be three balls left in the bag, B, C
and D for the second stage or draw. If the second ball drawn is B, the first
outcome constituting two balls is AB which is numbered 1 on the right most side
of the diagram 1.
If the first ball selected is B
and not replaced back, there will be three balls left in the bag, A, C and D.
At the second stage or draw, there are three possibilities, one of B, C or D
balls for second stage or draw. If the second ball drawn is A, the outcome
constituting two balls is BA which is numbered 4 on the right most side of the
diagram 1. Because they are different outcomes as the
position of the ball first or second is meaningful. Thus,
there are 12 permutations each with two balls selected randomly without replacement
from four balls.
Now, I will let you brainstorm how many will be the permutation of outcomes
without replacement if there are six balls out of which two balls are drawn one
at a time without 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 or object 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 not, wait
for my other statistical notes.
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