FINDING PROBABILITY USING A LIST

We all know, how to calculate theoretical probability using tables and tree diagrams.  

In this section, we are going to see, how theoretical probability can be calculated using lists. 

The formula given below can be used to find the theoretical probability.

Example 1 :

The combination for Khiem’s locker is a 3-digit code that uses the numbers 1, 2, and 3. Any of these numbers may be repeated. Find the probability that Khiem’s randomly-assigned number is 222.

Answer :

Step 1 :

List all the codes that start with 1 and have 1 as a second digit.

(1, 1, 1)

(1, 1, 2)

(1, 1, 3)

Step 2 :

List all the codes that start with 1 and have 2 as a second digit.

(1, 2, 1)

(1, 2, 2)

(1, 2, 3)

Step 3 :

List all the codes that start with 1 and have 3 as a second digit.

(1, 3, 1)

(1, 3, 2)

(1, 3, 3)

Step 4 :

We have now listed all the codes that start with 1. Let us repeat Steps 1–3 for codes that start with 2, and then for codes that start with 3. 

Step 5 :

Find the number of outcomes in the sample space by counting all the possible codes.

There are 27 such codes.

Step 6 :

Find the probability that Khiem’s locker code is 222.

Number of outcomes for  Khiem’s locker code 222  =  1

Probability (Code 222)  =  1/27

Example 2 :

Martha types a 4-digit code into a keypad to unlock her car doors. The code uses the numbers 1 and 0. If the digits are selected at random, what is the probability of getting a code with exactly two 0's ?

Answer :

Step 1 :

Total number of codes can be generated :

=  2⁴

=  16

Step 2 :

Let us consider the codes where we have exactly two 0's.

1100

0011

1010

0101

1001

0110

So, there are 6 codes where we get exactly two 0's.

Step 3 :

Let A be the event of getting a code with exactly two 0's. 

P(A)  =  6/64

P(A)  =  3/8

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