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.)
27x27x27x10x10x10
To find the probability of getting a license plate with a repeated letter or digit, we need to consider the total number of possible license plates and the number of license plates that have a repeated letter or digit.
First, let's calculate the total number of possible license plates. We know that the license plates have one numeral followed by three letters followed by three numerals.
For the first numeral, we have 10 options (0-9).
For the three letters, we have 26 options for each letter (A-Z), and since repetition is allowed, we have a total of 26^3 = 17,576 possibilities (26 * 26 * 26).
For the three numerals, we have 10 options for each numeral, resulting in 10^3 = 1,000 possibilities (10 * 10 * 10).
To calculate the total number of possible license plates, we multiply the number of choices for each section together: 10 * 17,576 * 1,000 = 175,760,000.
Now let's calculate the number of license plates with a repeated letter or digit.
To have a repeated letter in the three letters section, we can choose one of the 26 letters in 26 ways. The remaining two letters can be chosen from any of the 26 letters as well, so we have a total of 26^2 = 676 possibilities.
To have a repeated numeral in the three numerals section, we can choose one of the 10 numerals in 10 ways. The remaining two numerals can be chosen from any of the 10 numerals as well, so we have a total of 10^2 = 100 possibilities.
Finally, to calculate the total number of license plates with a repeated letter or digit, we multiply the number of choices for each section together: 26 * 676 * 10 * 100 = 1,768,000.
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)
Probability = 1,768,000 / 175,760,000
Calculating this value gives us a probability of 0.01, which means there is a 1% chance of getting a license plate with a repeated letter or digit.