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 combination 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 ‘Combination’ used in identifying the total number
of possible outcomes given the condition and calculating the probability of
events. I show the combination of total number outcomes using the tree diagram
1.
Diagram 1: Combination 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 same as the outcome
AB if the position of the ball whether the first or second is not meaningful.
Thus, the outcome BA is also numbered 2 in the diagram. Thus, there are six combinations
each with two balls selected randomly with replacement from three balls.
Now, I will let you brainstorm how many will be the 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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