Question

Telephone Numbers The telephone numbers for a region of Pennsylvania have an area code of 570. The next seven digits represent the local telephone numbers for that region. These cannot begin with a 0 or 1. In Exercises 15 and 16, assume your cousin lives within the given area code.
16. What is the probability of not randomly generating your cousin's telephone number on the first try?

Accepted Answer

Step 1: Understand the structure of the telephone number. The area code is fixed as 570, and the local telephone number consists of 7 digits. The first digit of the local number cannot be 0 or 1, leaving 8 possible choices (2 through 9). The remaining 6 digits can each be any digit from 0 to 9, giving 10 choices per digit. Step 2: Calculate the total number of possible telephone numbers in the region. Multiply the number of choices for the first digit (8) by the number of choices for each of the remaining 6 digits $$(10^6). $$This gives the total number of valid telephone numbers. Step 3: Recognize that your cousin has exactly one specific telephone number. Therefore, the probability of randomly generating your cousin's number on the first try is the reciprocal of the total number of valid telephone numbers, which is 1 divided by the total calculated in Step 2. Step 4: To find the probability of NOT randomly generating your cousin's telephone number on the first try, subtract the probability of generating their number from 1. This is expressed as: P(not cousin's number) = 1 - P(cousin's number). Step 5: Substitute the probability of generating your cousin's number (from Step 3) into the formula from Step 4 to compute the final probability of not generating your cousin's number on the first try.

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주요 개념

Probability

Sample Space

Complementary Events