the probability of collision

Question

Given a table (database) with 100 rows (slots). The row in the table is selected randomly every time new data are written to the table. What is the probability that on 10th time, a collision will occur? Collision is an instance when data is written to the place that already contains previously written data.

Answer 1

I assume this question is being asked in reference to a hashing function? Such problems can also be thought of as the birthday problem according to here. This website also discusses the problem of calculating hash collision probabilities -- it uses a Poisson approximation to estimate the value although an exact expression can always be found, it just may be difficult to evaluate. I will add more links to solving the birthday problem and hopefully some will help you (1)(2)(3).

Feller Volume 1 has a discussion of the birthday problem, with examples and tables demonstrating the accuracy of the Poisson approximation to the actual value.

$$1 - \left(\frac{1}{100}\right)^{10} \left(100\cdot 99 \cdot 98 \cdot 97 \cdot 96 \cdot 95 \cdot 94 \cdot93 \cdot 92 \cdot91 \right) $$

EDIT: Ball and bin problems usually assume there are more balls than bins, rather than fewer, whereas the birthday problem usually assumes fewer bins than balls (i.e. fewer people than days in the year, so the birthday problem is probably better to look at here.)

I believe this can also be rephrased as a balls and bins problem, of this type:

• another balls and bins question,
• If n balls are thrown into k bins, what is the probability that every bin gets at least one ball?
• Probability that all bins contain strictly more than one ball?
  In other words, the probability of a collision is the probability that one of the bins has more than one ball after the 10th round.