Questions: You pick 2 digits (0-9) at random without replacement, and write them in the order picked. What is the probability that you have written the first 2 digits of your phone number? Assume there are no repeats of digits in your phone number. Give your answer as a fraction.

Solution Steps

To solve this problem, we need to determine the probability of picking the first two digits of a phone number correctly when picking two digits at random without replacement.

1. Total Possible Outcomes: Since we are picking 2 digits out of 10 (0-9) without replacement, the total number of possible outcomes is the number of permutations of 10 digits taken 2 at a time.
2. Favorable Outcomes: There is only one specific sequence of 2 digits that matches the first two digits of the phone number.
3. Probability Calculation: The probability is the ratio of the number of favorable outcomes to the total number of possible outcomes.

The total number of possible outcomes when picking 2 digits out of 10 (0-9) without replacement is given by the number of permutations of 10 digits taken 2 at a time: \[ P(10, 2) = 10 \times 9 = 90 \]

There is only one specific sequence of 2 digits that matches the first two digits of the phone number. Therefore, the number of favorable outcomes is: \[ 1 \]

The probability is the ratio of the number of favorable outcomes to the total number of possible outcomes: \[ \text{Probability} = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{1}{90} \]

Final Answer