User Test Selection

Solution & Explanation

Problem Breakdown

• Population: 1,000,000 users
• Daily Test Selection: 1,000 users
• Selection Process: Random, with replacement

We need to determine:

1. Average Wait Time before a user is selected.
2. Probability of a user being chosen on the first day.

Understanding the Problem

The problem can be modeled using a geometric distribution because:

• We are interested in the number of trials (days) until the first success (user selection).
• Each day, the probability of a user being selected is constant.

Probability of Selection Each Day (p):

$p = \frac{1000}{1,000,000} = 0.001$

1. Average Wait Time Before Selection

For a geometric distribution with probability $p$, the expected number of trials (days) until the first success is given by:

$E[X] = \frac{1}{p} = \frac{1}{0.001} = 1000 \text{ days}$

Thus, on average, a user will wait 1000 days before being selected for the test.

2. Probability of Being Chosen on the First Day

The probability that a user is selected on the first day is simply the probability of selection $p$:

$P(\text{selected on first day}) = 0.001$

Since this probability is quite small (0.001), it is closer to zero than to one.

Conclusion

• Average Wait Time: 1000 days
• Probability of First-Day Selection: 0.001, which is closer to zero.

Additional Considerations

• Repetition of Selection: Since users can be selected more than once, this affects the distribution of selections over time, but not the individual probability each day.
• Geometric Distribution Properties: This distribution is appropriate here because it models the "waiting time" for the first occurrence of an event with a constant probability.