What is the probability that an 8 digit numbers contains "97"?
if the first 2 digits are 97 then there are 10^6 ways of picking the other 6 digits.
Since the 97 can appear in 7 different places in an 8-digit number, that means there are 7,000,000 ways to make an 8-digit numbers with 97 in it.
so, that would be a probability of 7*10^6/9*10^7 = 7/90 = .0777
To determine the probability that an 8-digit number contains "97", we first need to find the total number of possible 8-digit numbers and then determine the number of those numbers that contain "97".
Step 1: Total Number of 8-Digit Numbers
An 8-digit number can range from 1,000,000 to 9,999,999 since it must have 8 digits. Therefore, there are a total of (9,999,999 - 1,000,000 + 1) = 9,000,000 possible combinations.
Step 2: Number of 8-Digit Numbers with "97"
To find the number of 8-digit numbers that contain "97", we can look specifically at the placement of "97" in the sequence. Since "97" is a 2-digit number, it can appear in six different positions within the 8-digit number:
1. The number starts with "97" (e.g. 97000000).
2. The second and third digits are "97" (e.g. 90790000).
3. The third and fourth digits are "97" (e.g. 90097000).
4. The fourth and fifth digits are "97" (e.g. 90009700).
5. The fifth and sixth digits are "97" (e.g. 90000970).
6. The number ends with "97" (e.g. 90000097).
For each of these six positions, the remaining six digits can be any number from 0 to 9 (except for "97" in that position). Therefore, the number of 8-digit numbers that contain "97" is 6 * 10^6 (six positions times ten choices for each of the remaining digits).
Step 3: Calculate the Probability
Finally, we can calculate the probability by dividing the number of 8-digit numbers that contain "97" by the total number of 8-digit numbers:
Probability = (Number of 8-Digit Numbers with "97") / (Total Number of 8-Digit Numbers)
Probability = (6 * 10^6) / 9,000,000
Probability = 2/3
Therefore, the probability that an 8-digit number contains "97" is 2/3 or approximately 0.6667.