Question

a three - digit personal identification number is chosen using the digits 1 - 9. the digits cannot be repeated. what is the approximate probability that the first digit will be 6? 0.00595 0.03571 0.11111 0.66667

Explanation:

Step1: Calculate the total number of three - digit PINs

The number of permutations of \(n\) objects taken \(r\) at a time is \(P(n,r)=\frac{n!}{(n - r)!}\). Here \(n = 9\) (digits \(1-9\)) and \(r=3\). So \(P(9,3)=\frac{9!}{(9 - 3)!}=\frac{9!}{6!}=9\times8\times7 = 504\).

Step2: Calculate the number of three - digit PINs with first digit 6

If the first digit is 6, then we need to choose 2 digits from the remaining 8 digits. Using the permutation formula \(P(8,2)=\frac{8!}{(8 - 2)!}=\frac{8!}{6!}=8\times7=56\).

Step3: Calculate the probability

Probability \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\). So \(P=\frac{56}{504}=\frac{1}{9}\approx0.11111\).

Answer:

\(0.11111\) (the third option)