What is the probability of getting a license plate that has a repeated letter or digit if you live in a state that has three numerals followed by five letters followed by one numeral? (Round to the nearest whole percent.)
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 that satisfy the condition.
In this case, we have a license plate format with three numerals followed by five letters followed by one numeral.
Step 1: Calculate the total number of possible license plates.
For the numerals, we have 10 possible options (0-9). The same goes for the single numeral at the end. So, the total number of options for numerals is 10 * 10 = 100.
For the letters, we have 26 options (A-Z) and since we have five letters, the total number of options for letters is 26^5 = 11,881,376.
To calculate the total number of possible license plates, we multiply these values together: 100 * 11,881,376 = 1,188,137,600.
Step 2: Determine the number of license plates with repeated letters or digits.
To have a repeated letter or digit on the license plate, there are several different cases:
- One numeral repeated in the first three positions.
- One numeral repeated in the last two positions.
- One letter repeated in two positions.
- One letter repeated in three positions.
- One letter repeated in four positions.
- One numeral repeated in the last position.
To calculate the number of license plates with repeated letters or digits, we need to consider each case separately and then sum them up.
Case 1: One numeral repeated in the first three positions
- We have 10 choices for the repeated numeral (0-9).
- There are 26 options for each of the five letters.
- We have 10 choices for the last numeral.
- So, the number of license plates with a repeated numeral in the first three positions is 10 * 26^5 * 10 = 1,873,760.
Case 2: One numeral repeated in the last two positions
- We have 10 choices for the first numeral.
- There are 26 options for each of the five letters.
- We have 10 choices for the repeated numeral.
- So, the number of license plates with a repeated numeral in the last two positions is 10 * 26^5 * 10 = 1,873,760.
Case 3: One letter repeated in two positions
- We have 10 choices for the first numeral.
- There are 26 options for each of the four remaining unique letters.
- We have 26 choices for the repeated letter.
- We have 10 choices for the last numeral.
- So, the number of license plates with a repeated letter in two positions is 10 * 26^4 * 26 * 10 = 704,960,000.
Cases 4, 5, and 6 follow a similar logic as Cases 3 and result in the same number of license plates: 704,960,000.
Thus, the total number of license plates with repeated letters or digits is: (1,873,760 + 1,873,760 + 704,960,000 + 704,960,000 + 704,960,000) = 2,114,327,520.
Step 3: Calculate the probability of getting a license plate with a repeated letter or digit.
The probability is given by the ratio of the number of license plates with repeated letters or digits to the total number of license plates:
Probability = (Number of license plates with repeated letters or digits) / (Total number of possible license plates)
Probability = 2,114,327,520 / 1,188,137,600
Probability = 1.780 (rounded to three decimal places)
So, the probability of getting a license plate that has a repeated letter or digit in this scenario is approximately 1.780% when rounded to the nearest whole percent.
To find the probability of getting a license plate that has a repeated letter or digit, we can consider the total number of possible license plates and the number of license plates that have repeated letters or digits.
First, let's calculate the total number of possible license plates.
Since the state has three numerals followed by five letters followed by one numeral, we have the following possibilities:
3 numerals: 10 options (0-9)
5 letters: 26 options (A-Z)
1 numeral: 10 options (0-9)
Thus, the total number of possible license plates is 10 × 10 × 10 × 26 × 26 × 26 × 26 × 26 × 10 = 26^5 × 10^3.
Next, let's calculate the number of license plates that have repeated letters or digits.
There are two cases to consider: repeated letters and repeated digits.
Case 1: Repeated Letters
We have 26 options for each letter spot, and since there are five letter spots, we have 26^5 possibilities.
Case 2: Repeated Digits
We have 10 options for each numeral spot, and since there are three numeral spots, we have 10^3 possibilities.
To find the total number of license plates with repeated letters or digits, we sum the two cases:
Total = 26^5 + 10^3
Now, we can find the probability by dividing the number of license plates with repeated letters or digits by the total number of possible license plates:
Probability = (Total / Total Number of Possible License Plates) × 100
Plugging in the values and calculating, we get:
Probability = (26^5 + 10^3) / (26^5 × 10^3) × 100
Rounding to the nearest whole percent, the probability of getting a license plate that has a repeated letter or digit is approximately 0.17%.
number of all possible plates
=10^3 * 26^5 * 10
= 118813760000 (this includes all types of repetition)
number of plates without any repetition
= 10*9*8*26*25*24*23*22*7
3.9783744 x 10^10
so prob of some kind of repeat
= 1 - 3.9783744 x 10^10/118813760000
= .665