Suppose that Social Security numbers were issued uniformly...

Suppose that Social Security numbers were issued uniformly at random, with replacement. That is, your Social Security number would consist of just nine randomly generated digits, and no check would be made to ensure that the same number was not issued twice. Sometimes, the last four digits of a Social Security number are used as a password. How many people would you need to have in a room before it was more likely than not that two had the same last four digits? How many numbers could be issued before it would be more likely than not that there is a duplicate number? How would you answer these two questions if Social Security numbers had 13 digits? Try to give exact numerical answers.

Key Concepts

Birthday Problem

This is the classic probability puzzle that asks for the minimum number of independently chosen items (such as people’s birthdays or digits) needed so that the probability of at least one duplicate exceeds a certain threshold (usually 50%). The problem illustrates that even when the total number of possibilities is large, the number of samples required for a likely repeat is much smaller than one might intuitively expect.

Uniform Random Sampling

This concept assumes that every possible outcome, such as each Social Security number or a block of its digits, is equally likely and that each draw is independent (with replacement). It provides the basis for modeling the problem mathematically by assigning equal probabilities to all outcomes, which is an essential assumption in the birthday problem and collision probability calculations.

Collision Probability Calculation

In problems involving duplicate numbers, the key quantity is the probability that all values drawn from a large set are distinct. This is typically computed as the product of terms that account for the decreasing number of available distinct choices. The problem then finds the threshold number of draws for which this probability falls below a desired level (for instance, below 50%), thereby indicating that a collision is more likely than not.

Exponential Approximation in Large Sample Problems

When the number of possibilities is large, the product used in the exact calculation of collision probability becomes unwieldy. An effective approximation is to use the exponential function – by comparing the product with an exponential expression whose exponent is related to the number of pairs drawn (approximately n(n?1)/(2N)) – to estimate the threshold number of samples. This is the basis for the common approximation n ? ?(2N ln 2) used in the birthday problem.

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