What is the probability of getting a license plate that has a repeated letter or digit if you live in a state that has one numeral followed by two letters followed by one numeral? (Round to the nearest whole percent.)
To calculate 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 then find the number of license plates that satisfy the given condition.
In this case, we have a license plate structure where the first and last characters are numerals, and the middle two characters are letters. Let's break down the process step by step.
1. Determine the number of possible numerals for the first and last characters: Since we are given one numeral for both the first and last characters, we have 10 options for each position (0-9).
2. Determine the number of possible letters for the middle two characters: Since we are given two letters for the middle positions, we have 26 options for each position (A-Z).
3. Calculate the total number of possible license plates: Multiply the number of options for each character together: 10 options for the first numeral, 26 options for the first letter, 26 options for the second letter, and 10 options for the last numeral. So, the total number of possible license plates is 10 * 26 * 26 * 10 = 676,000.
4. Calculate the number of license plates with repeated letters or digits:
- Case 1: Repeated letter in the middle two characters: We have 26 options for the first letter and 1 option for the second letter, as it must be the same as the first letter. Additionally, we have 10 options for both the first and last numerals. So, the number of license plates with a repeated letter in the middle two characters is 26 * 1 * 10 * 10 = 2,600.
- Case 2: Repeated digit in the middle two characters: We have 26 options for each of the two letters, so the number of possible combinations is 26 * 26 = 676. Within these combinations, there are 10 options for both the first and last numerals. The repeated digit can be placed in either the first or second position, so we multiply the total by 2. So, the number of license plates with a repeated digit in the middle two characters is 676 * 10 * 10 * 2 = 135,200.
5. Calculate the total number of license plates with a repeated letter or digit: Add the results from both cases: 2,600 + 135,200 = 137,800.
6. Calculate the probability: Divide the total number of license plates with a repeated letter or digit by the total number of possible license plates, and then multiply by 100 to get a percentage: (137,800 / 676,000) * 100 = 20.36%.
Therefore, the probability of getting a license plate with a repeated letter or digit, rounded to the nearest whole percent, is 20%.