Question

question 16 · 1 point
mary needs to create a 4 - digit numerical pin for her bank card. if the digits are chosen randomly, what is the probability that her pin does not have any repeating digits?
express your answer as a decimal rounded to three decimal places.
provide your answer below:
p ≈ □

Explanation:

Step1: Find total possible PINs

A 4 - digit PIN, each digit can be 0 - 9 (10 choices). So total number of possible PINs is \(10\times10\times10\times10 = 10^4=10000\) (by the multiplication principle, as each of the 4 positions has 10 options).

Step2: Find PINs with no repeating digits

For the first digit, we have 10 choices (0 - 9). For the second digit, we can't use the digit we used for the first digit, so we have 9 choices. For the third digit, we can't use the first two digits, so we have 8 choices. For the fourth digit, we can't use the first three digits, so we have 7 choices. By the multiplication principle, the number of 4 - digit PINs with no repeating digits is \(10\times9\times8\times7=\frac{10!}{(10 - 4)!}=5040\).

Step3: Calculate the probability

The probability \(P\) that a PIN has no repeating digits is the number of favorable outcomes (no repeating digits) divided by the number of total outcomes. So \(P=\frac{10\times9\times8\times7}{10\times10\times10\times10}=\frac{5040}{10000}=0.504\).

Answer:

\(0.504\)