Question

What is the probability getting a license plate that has a repeated letter or digit if you live in a state in which license plate have two letters followed by four numerals? (round your answer to one decimal place)

Answer 1

What is the probability getting a license plate that has a repeated letter or digit if you live in a state in which license plate have two letters followed by four numerals?

Solution :

Now to answer this question we first calculate the total number of license plate possible with the given condition .

So given in the state   license plate have two letters followed by four numerals .

So there are 6 positions in a    license plate that have two letters followed by four numerals .

Now each of these 2 letters can be any of 26 letters , similarly   each of these 4 digits   can be any of 10 numerals .

So for any random license in the state it's 6 position can be occupied in the following ways :

1st position : have 26 choices ( 26 letters of alphabet )

2nd position : have 26 choices ( 26 letters of alphabet )

3rd position : have 10 choices (10 digits of numerals )

4th position : have 10 choices ( 10 digits of numerals )

5th position : have 10 choices (10 digits of numerals )

6th position : have 10 choices ( 10 digits of numerals )

So by multiplication principle :

the total number of license plate possible    in the state which  have two letters followed by four numerals are :

Now we wish to   calculate probability of  getting a license plate that has a repeated letter or digit .

Now complement of this event is :

calculating   probability of  getting a license plate that do not have any repeated letter or any repeated digit .

And we know

So  

probability of  getting a license plate that has a repeated letter or digit

   probability of  getting a license plate that do not have any repeated letter or any repeated digit .

So let us calculate total number of license plate possible that do not have any repeated letter or digit  

So there are 6 positions in a    license plate that have two letters followed by four numerals .

we don't want any repeated letter or any repeated digit   . .

So here in this case 6 position can be occupied in the following ways :

1st position : have 26 choices ( 26 letters of alphabet )

2nd position : have 25 choices ( because we have used 1 of the letter in 1st position and letters cannot be repeated , hence we are left with remaining 25 letters of alphabet   )

3rd position : have 10 choices (10 digits of numerals )

4th position : have 9 choices ( because we have used 1 of the numerals in 3rd position and digits  cannot be repeated , hence we are left with remaining 9 numerals   )

5th position : have 8 choices ( because we have used 2  numerals in 3rd and 4th position )

6th position : have 7 choices ( because we have used 3  numerals in 3rd , 4th and 5th position )  

So by multiplication principle :

the   total number of license plate possible that do not have any repeated letter or any repeated digit   are :

Therefore we get   probability of  getting a license plate that do not have any repeated letter or digit . as :

So probability of  getting a license plate that do not have any repeated letter or any repeated digit . is :

Hence  

probability of  getting a license plate that has a repeated letter or digit

   probability of  getting a license plate that do not have any repeated letter or digit .

probability of  getting a license plate that has a repeated letter or digit   

.............. ( rounded to one decimal place )

Therefore answer is :

the probability getting a license plate that has a repeated letter or digit if you live in a state in which license plate have two letters followed by four numerals is :