Question

you are to create a 4 - digit passcode for your phone. the first digit cannot be a zero or a 1. the 2nd digit must be ≥ 7. the 3rd digit must be less than 7. the 4th digit is either a zero or a 1. what is the probability that the 2 middle digits of your pass code will be 70? a) $\frac{18}{336}$ b) $\frac{16}{336}$ c) $\frac{16}{338}$ d) $\frac{18}{896}$

Explanation:

Step1: Calculate number of choices for each digit

The first digit cannot be 0 or 1, so there are 8 choices (2 - 9). The second - digit must be $\geq7$, so there are 3 choices (7, 8, 9). The third - digit must be less than 7, so there are 7 choices (0 - 6). The fourth - digit is either 0 or 1, so there are 2 choices. The total number of possible 4 - digit passcodes is $8\times3\times7\times2=336$ by the multiplication principle.

Step2: Calculate number of passcodes with middle digits 70

The first digit still has 8 choices (2 - 9). The second digit is fixed as 7 (1 choice), the third digit is fixed as 0 (1 choice), and the fourth digit has 2 choices (0 or 1). So the number of passcodes with middle digits 70 is $8\times1\times1\times2 = 16$.

Step3: Calculate the probability

The probability $P$ that the two middle digits are 70 is the number of favorable outcomes divided by the number of total outcomes. So $P=\frac{16}{336}$.

Answer:

B. $\frac{16}{336}$