Telephone company estimates that after one year they will have up to 100 000 users, and after five years, there will be 250 000 users. Each user is assigned a randomly generated 3,4, or 5 digit number. To ensure the password is difficult to guess the digits cannot all be the same ( 111, or 2222 cannot be used).

The total number of passwords that can be generated using these restrictions is 110 970. The total number of 3 digit passwords is 990, the number of 4 digit passwords is 9990 and the number of 5 digit passwords is 99 990.



Telephone company has done an initial study to determine the probability that users will forget their password. The experimental probabilities are given in the table.

Number of Characters Probability of Forgetting

3... 0.0001

4... 0.00015

5... 0.0002


On any given day, 10 000 of your customers are expected to use this voicemail system. Determine the probability that at least one customer will forget their password

a) 3 characters
b) 4 chatacters
c) 5 characters


We have been taught that in an 'at least' question to determine the none first. So in this case we have to figure out the probability of NO ONE forgetting their password before we can figure out at least one. So from this information how do I figure out the probability of no users forgetting their passwords?

so the probability of any one user not forgetting his password (for 3 number) is

1-0.0001 = 0.9999

now the probability of none of 10000 people forgetting their password is

0.9999^10000 = 0.368

so the probability of at least one person (not none) forgetting his password is 1-0.368

Now you can do the same for b and c.

Why is it 0.368^10000?

Er, sorry why is do you put it to the power of 10 000?

the probability of one user forgetting their password is 0.0001

the probability of 2 users forgetting is 0.0001*0.0001 = 0.0001^2

the probability of 5 users forgetting their password is 0.. 0001*0.0001*0.0001*0.0001*0.0001 = 0.0001^5

etc.

the same applies for the number of people not forgetting their password

so the chance of 10000 people not forgetting their password is 0.9999^10000

So then I just do the same thing for b and c.


When I do I get 0.777 for b and 0.865 for c. Is that right?

Sounds about right to me.

Thank you So much! You have no idea how much help you have been!

You're very welcome, you know you can rate me by clicking *rate this answer* if I have been helpful :)