What is the probability of getting a license plate that has a repeated letter or digit if you live in a state that has four letters followed by two numerals followed by two letters? (Round to the nearest whole percent.)
The licence plate template is
XXXXNNXX
with 6 letters and two numbers.
The number of licence plates without repetition, i.e. without choosing a previously used character or number is
=26.25.24.23.10.9.22.21
=C(26,6)*C(10,2)
The number of licence plates without regard to repetition is
=26^6*10*2
The probability P1 of getting a licence plate without repetition is the first divided by the second.
The probability of getting a plate with repetition is 1-P1.
Sorry,
26.25.24.23.10.9.22.21
is not C(26,6)*C(10,2)
The rest of the calculations should be good.
looks like this is the day for errors
I left out one of the letters in the first group of letters
should have said:
so what we don't want are cases where all letters and numbers are different, which would be
26*25*24*23*10*9*22*21 = 14 918 904 000
so the prob as asked in your question
= 1 - 14 918 904 000/30 891 577 600
= .517
Well, when it comes to license plates, it can feel like you're playing a game of "Wheel of Fortune." So let's spin that imaginary wheel and see what we get! In this case, we're looking for a repeated letter or digit.
To calculate the probability, we need to figure out how many options we have for each segment.
For the first segment of four letters, we have 26 options (A-Z) for each letter. So that's 26^4.
For the second segment of two numerals, we have 10 options (0-9) for each number. So that's 10^2.
Lastly, for the final two letter segment, we also have 26 options (A-Z) for each letter. So that's 26^2.
Now, to determine the number of license plates with repeated letters or digits, we need to subtract the number of license plates with no repeated letters or digits from the total number of possible license plates.
The formula becomes: (total number of possible plates) - (number of plates with no repeats).
Total number of possible plates = 26^4 * 10^2 * 26^2
Now, let's calculate the number of plates with no repeats.
For the first segment of four letters, we have 26 options (A-Z) for each letter. However, since we can't repeat any letter, after choosing the first letter we have 25 options, then 24 options for the next letter, then 23 options for the third letter, and finally 22 options for the fourth letter. So that's 26 * 25 * 24 * 23.
For the second segment of two numerals, we have 10 options (0-9) for the first number, and then 9 options for the second number (since we can't repeat). So that's 10 * 9.
Lastly, for the final two letter segment, we also have 26 options (A-Z) for the first letter, and then 25 options for the second letter (since we can't repeat). So that's 26 * 25.
Now, let's calculate the number of license plates with no repeats: 26 * 25 * 24 * 23 * 10 * 9 * 26 * 25.
To find the number of license plates with repeated letters or digits, we subtract the number of plates with no repeats from the total number of possible plates:
(26^4 * 10^2 * 26^2) - (26 * 25 * 24 * 23 * 10 * 9 * 26 * 25).
Now, let's divide the number of plates with repeated letters or digits by the total number of possible plates, and multiply by 100 to get a percentage.
And voila! The probability of getting a license plate with a repeated letter or digit is the result of that calculation, rounded to the nearest whole percent. I'll let you plug in the numbers and do the math. Good luck!
To find the probability of getting a license plate with a repeated letter or digit, we need to calculate the total number of possible license plates and the number of license plates that have repeated letters or digits.
In this case, we can break down the license plate format as follows:
1. Four letters: There are 26 letters in the English alphabet, so there are 26 choices for each of the four letters. Thus, the total number of possibilities for the four letters is 26^4 = 456,976.
2. Two numerals: There are 10 digits (0-9), so there are 10 choices for each of the two numerals. Therefore, the total number of possibilities for the numerals is 10^2 = 100.
3. Two letters: As mentioned before, there are 26 choices for each letter. So the total number of possibilities for the two letters is 26^2 = 676.
To find the total number of possible license plates, we multiply the possibilities for each section: 456,976 (letters) * 100 (numerals) * 676 (letters) = 30,778,176,000.
Now let's calculate the number of license plates with repeated letters or digits.
1. Repeated letters: There are 26 letters to choose from and 4 slots for letters. So the number of license plates with repeated letters is 26 * 26 * 26 * 26 = 456,976.
2. Repeated digits: There are 10 digits to choose from and 2 slots for numerals. So the number of license plates with repeated digits is 10 * 10 = 100.
To find the number of license plates with repeated letters or digits, we sum these two possibilities: 456,976 (repeated letters) + 100 (repeated digits) = 457,076.
Finally, we can calculate the probability of getting a license plate with a repeated letter or digit by dividing the number of license plates with repeated letters or digits by the total number of possible license plates and multiplying by 100 to obtain a percentage:
(457,076 / 30,778,176,000) * 100 ≈ 0.00148%
Therefore, the probability of getting a license plate that has a repeated letter or digit is approximately 0.00148%.
number of total plates including all possible repetitions
= 26^4*100*26^2 = 30 891 577 600 (wow, enough until cars become "extinct")
so what we don't want are cases where all letters and numbers are different, which would be
26*25*24*10*9*23*22 = 710 424 000
so prob with some kind of repeats
= 1 - .710 424 000/30 891 577 600
= appr .9777 or 98% to the nearest percent