The serial number on a $10 bill contains eight non-zero digits. If you $10 bill contains the digit 7 at least once, you win a prize. The probability that your $10 bill will be a prize-winning bill is?
Without a zero, there must be 9^8 number of possible digits. Further there must be 8^8 possible digits that exclude both zero and 7. So, the probability at at least one seven appears is 1.-((8^8)/(9^8))
To find the probability of winning a prize with the given criteria, we need to determine the total possible combinations of eight non-zero digits on a $10 bill and the number of combinations that contain at least one digit 7.
Let's break down the problem step by step:
Step 1: Total possible combinations
Since we have eight non-zero digits, each digit can take any value from 1 to 9. Thus, there are 9 possible choices for each digit. Therefore, the total possible combinations of eight non-zero digits on a $10 bill can be calculated as 9^8 (since we have 8 digits to fill).
Total possible combinations = 9^8 = 43,046,721
Step 2: Combinations with at least one digit 7
Now, let's calculate the number of combinations that contain at least one digit 7.
To find this, we can calculate the total number of combinations without any 7's and subtract it from the total number of possible combinations.
To calculate the combinations without any 7's:
Since 7 is omitted, each digit can take any value from 1 to 6 and 8 to 9. Thus, there are 8 possible choices for each digit. Therefore, the combinations without any 7's can be calculated as 8^8 (since we have 8 digits to fill).
Combinations without any 7's = 8^8 = 16,777,216
To find the combinations with at least one digit 7:
Combinations with at least one digit 7 = Total possible combinations - Combinations without any 7's
Combinations with at least one digit 7 = 43,046,721 - 16,777,216 = 26,269,505
Step 3: Calculating the probability
Finally, we can calculate the probability of winning a prize by dividing the number of combinations with at least one digit 7 by the total possible combinations.
Probability = Combinations with at least one digit 7 / Total possible combinations
Probability = 26,269,505 / 43,046,721
Calculating the probability approximately, we get:
Probability ≈ 0.6102 or 61.02%
Therefore, the probability that your $10 bill will be a prize-winning bill is approximately 0.6102 or 61.02%.