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Chapter 4: Problem 6

Express all probabilities as fractions. A Social Security number consists of nine digits in a particular order, and repetition of digits is allowed. After seeing the last four digits printed on a receipt, if you randomly select the other digits, what is the probability of getting the correct Social Security number of the person who was given the receipt?

Short Answer

Expert verified
The probability is \(\frac{1}{100000}\).

Step by step solution

01

- Understand the Problem Statement

A Social Security number (SSN) is a nine-digit number where each digit can be anything from 0 to 9 and repetition is allowed. We need to find the probability of correctly guessing the entire SSN if the last four digits are already known.
02

- Determine Possible Digits

Since each digit can be anything from 0 to 9, there are 10 possible digits for each of the unknown five positions in the SSN.
03

- Calculate Total Possible Combinations

The total number of possible combinations for the five unknown digits can be calculated using the formula for permutations with repetition, which is given by \[ 10^5 \] This is because each of the five positions can be any of the 10 digits.
04

- Calculate the Probability of the Correct Combination

Since there is only one correct combination out of the total possible combinations, the probability of guessing it correctly is the ratio of the number of successful outcomes to the total number of possible outcomes.\[ P(\text{correct combination}) = \frac{1}{10^5} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Social Security number
A Social Security number (SSN) in the United States is a unique identifier assigned to individuals. It consists of nine digits, and each digit can be any number from 0 to 9. For example, an SSN might look like 123-45-6789. The digits are assigned in a particular order and can be repeated, which gives rise to numerous possible combinations.
Understanding the structure of SSNs is essential for solving probability problems related to them. In this specific exercise, knowing that there are nine digits with repetition allowed is key to calculating the total number of possible combinations of these digits.
Permutations with repetition
Permutations with repetition occur when we're forming ordered sequences (permutations) from a set of items, and repetition of items is allowed. This is an important concept because it significantly increases the number of possible combinations.
In the context of our Social Security number problem, we need to determine the number of possible SSN combinations for the five unknown digits. Since there are 10 possible digits (0 through 9) for each of the five positions, we use the following formula to calculate the total number of permutations with repetition:
gelate \((10^5)=100,\text{}000\).
This formula shows that each digit allows for 10 possibilities, repeated across 5 digits.
In general, if you have \(n\) positions and each position can be filled by one of \(k\) different items, there are \(k^n\) possible combinations.
Successful outcomes
In any probability problem, understanding successful outcomes is crucial. A successful outcome is one that meets the specific conditions set by the problem.
In the exercise given, the successful outcome is guessing the five unknown digits of the SSN correctly. Since there's exactly one correct combination out of all possible ones, we represent it as the fraction of 1 out of the total number of possible combinations, calculated earlier:
gelate \(\frac{1}{10^5}=0.00001.\).Simply put, only 1 out of 100,000 possible combinations is correct, making the probability of guessing the correct SSN if the other four digits are known very low.
Understanding this fraction helps you grasp the concept of probability and see how its value becomes smaller when the number of possible outcomes is large.

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Most popular questions from this chapter

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While this exercise was being created, Weather.com indicated that there was a \(60 \%\) chance of rain for the author's home region. Based on that report, which of the following is the most reasonable interpretation?a. \(60 \%\) of the author's region will get rain today. b. In the author's region, it will rain for \(60 \%\) of the day. c. There is a 0.60 probability that it will rain somewhere in the author's region at some point during the day.

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