Outcomes Without Replacement and Tree Diagram, Statistical Note 8

A bag contains four balls of different colors (A of Red color, B Blue, C Green and D yellow). Select the first ball at random and do not replace back to the bag. Select remaining balls one at a time following the same process. Calculate the total number of possible outcomes containing all four balls.

Calculating the total number of outcomes using all objects selected one at a time without replacement is important to calculate the probability of events. Also, important is a mathematical concept of the ‘Factorial’ used in identifying the total number of possible outcomes and calculating the probability of events. I show the total number outcomes using the tree diagram 1.

Diagram 1: Outcomes Without Replacement

There will be four stages required to arrange all four balls sequentially. 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. At the second stage or draw, there are three possibilities, one of B, C or D balls. If the second ball drawn is B and not replaced back, there will be only two balls left in the bag leaving two possibilities C or D for the third stage or draw. If the third ball drawn is green (C) and not replaced back, the fourth ball left is by default yellow (D). Now, the first outcome constituting all four balls is ABCD which is numbered 1 on the right most side of the diagram 1. The second outcome is ABDC, numbered 2 in the diagram. In the same way, there will be altogether 24 outcomes, each constituting four balls. Every outcome is unique in terms of position of four balls. Note that this uniqueness of an outcome because of the position of each of four balls is called 'Permutation'. We’ll discuss more about Permutation in some other notes.

The total number of possible outcomes is calculated by multiplying four possibilities at the first stage, three possibilities at the second stage, two possibilities at the third stage and one possibility at the fourth or last stage. This is calculated as 4 X 3 X 2 X 1 equal to 24 represented by the first number with exclamation symbol, denoted by 4!, which is termed as four factorial. Thus, the concept of factorial is used to calculate the number of possible outcomes given the number of balls, a case in which a ball is not replaced back once drawn. This could be termed as outcomes without replacement.

How many will be the outcomes without replacement if there are five balls? Can you calculate using the factorial formula? Now, you could use the formula 5!, which is 5 X 4 X 3 X 2 X 1 equal to 120. Can you show in the diagram? Perhaps, difficult to show.  

Thus, the diagrammatic logic is important to understand how to calculate the number possible outcomes without replacement. As the number of balls or entities increases, the diagram becomes complicated and it will be less possible to show all outcomes in the diagram. With this factorial formula, one can easily calculate the number of possible outcomes given the number of balls or entities.

Now, I will let you brainstorm what will be the possible number of outcomes using three balls drawn one at a time with replacement, a case in which a ball is replaced back once drawn before drawing the next ball. Could you use a diagram to show all possible outcomes? Could you use any formula? My next note 9 will discuss about it.