Question

If you had a 5 digit pin (numbers 0 - 9) with all combinations of digits being equally likely, what is the probability of guessing the pin if you try 3 times with a different pin each time?

Answer 1

If repetition is not allowed in the 5 digit pin, then number of possible pins = 10C5 = 252
P(guessing it right in the first attempt) = 1/252
P(guessing it right in the second attempt) = 251/252 * 1/251 = 1/252
P(guessing it right in the second attempt) = 251/252 * 250/251 * 1/250 = 1/252
P(of guessing it right in 3 attempts .with different pin each time) = 1/252 + 1/252 + 1/252 = 3/252 = 1/84 = 0.0119

If repetition is allowed in the 5 digit pin, then number of possible pins = 10^5 = 100000
P(guessing it right in the first attempt) = 1/100000
P(guessing it right in the second attempt) = 99999/100000 * 1/99999 = 1/100000
P(guessing it right in the second attempt) = 99999/100000 * 99998/99999 * 1/99998 = 1/100000
P(of guessing it right in 3 attempts .with different pin each time) = 1/100000 + 1/100000 + 1/100000 = 3/100000 = 0.00003