What is the probability of getting a license plate that has a repeated letter or digit if you live in a state in which license plates have one numeral followed by three letters followed by three numerals? (Round your answer to one decimal place.)
Probability of not repeating, P(no repeat)
=10*26*25*24*9*8*7 / (10^4 * 26^4)
Probability repeated
=1-P(no repeat)
To find the probability of getting a license plate with a repeated letter or digit, we need to determine the total number of possible license plates and the number of license plates with a repeated letter or digit.
First, let's calculate the total number of possible license plates.
Given that license plates have one numeral followed by three letters followed by three numerals, we can calculate the possibilities for each part separately.
For the numeral part, there are 10 possible digits (0-9).
For the letter part, there are 26 letters in the English alphabet.
For the last three numeral part, again we have 10 possible digits.
Therefore, the total number of possible license plates is calculated as:
Number of possibilities for numeral part x Number of possibilities for letter part x Number of possibilities for numeral part
10 x 26 x 26 x 26 x 10 x 10 x 10 = 17,576,000
Now, let's calculate the number of license plates with a repeated letter or digit.
Case 1: Repeated letter
To have a repeated letter, we must choose one letter from 26 possibilities for the letter part and place it in either the second or third position. The other two positions can be filled with any of the remaining 25 letters.
Number of possibilities with a repeated letter = Number of possibilities for numeral part x 26 x 25 x 25 x 10 x 10 x 10 = 65,000,000
Case 2: Repeated digit
To have a repeated digit, we must choose one digit from 10 possibilities for the numeral part and place it in either the fourth, fifth, or sixth position. The other two positions can be filled with any of the remaining 9 digits.
Number of possibilities with a repeated digit = 10 x 26 x 26 x 26 x 9 x 9 x 10 = 61,092,000
Finally, let's calculate the total number of license plates with a repeated letter or digit.
Total number of license plates with a repeated letter or digit = Number of possibilities with a repeated letter + Number of possibilities with a repeated digit
= 65,000,000 + 61,092,000 = 126,092,000
Now, to find the probability, we divide the number of license plates with a repeated letter or digit by the total number of possible license plates.
Probability = Number of license plates with a repeated letter or digit / Total number of possible license plates
= 126,092,000 / 17,576,000
Dividing these numbers gives the probability, which is approximately 7.2 (rounded to one decimal place).
Therefore, the probability of getting a license plate that has a repeated letter or digit is approximately 7.2%.