Question

a 4 - digit pin number is selected. what it the probability that there are no repeated digits? the probability that no numbers are repeated is. write your answer in decimal form, rounded to the nearest thousandth.

Explanation:

Step1: Calculate total number of 4 - digit PINs

Each digit of the PIN can be chosen from 0 - 9. So the total number of 4 - digit PINs is $10\times10\times10\times10=10^4 = 10000$.

Step2: Calculate number of 4 - digit PINs with no repeated digits

For the first digit, we have 10 choices (0 - 9). For the second digit, we have 9 choices (to avoid repeating the first - digit). For the third digit, we have 8 choices (to avoid the first two digits), and for the fourth digit, we have 7 choices (to avoid the first three digits). So the number of 4 - digit PINs with no repeated digits is $P(10,4)=\frac{10!}{(10 - 4)!}=10\times9\times8\times7 = 5040$.

Step3: Calculate the probability

The probability $P$ that there are no repeated digits is the number of favorable outcomes (PINs with no repeated digits) divided by the number of total outcomes (all 4 - digit PINs). So $P=\frac{5040}{10000}=0.504$.

Answer:

$0.504$