How many job applications until you get hired? Solved step by step

Problem. Suppose each job application has a 2% chance of turning into an offer, and applications are independent of each other. How many applications do you need so that the chance of at least one offer reaches 50%, and what is that chance after 100 applications?

Step by step

1. Each application fails with probability 1 − 0.02 = 0.98. Because the applications are independent, the chance that all n of them fail is 0.98^n.

2. 'At least one offer' is the opposite of 'all fail', so the chance of at least one offer is 1 − 0.98^n.

3. For a 50% chance, solve 1 − 0.98^n = 0.5, which gives 0.98^n = 0.5.

4. Take logarithms: n × ln 0.98 = ln 0.5, so n = ln 0.5 ÷ ln 0.98 ≈ (−0.6931) ÷ (−0.0202) ≈ 34.3.

5. Since n must be a whole number, round up to 35 applications. At 34 the chance is just under 50%, and at 35 it is just over.

6. For 100 applications: 0.98^100 ≈ 0.133, so the chance of at least one offer is 1 − 0.133 ≈ 0.867, about 87%.

7. For a 90% chance, solve 0.98^n = 0.1, so n = ln 0.1 ÷ ln 0.98 ≈ (−2.3026) ÷ (−0.0202) ≈ 114 applications.

Answer: The chance of at least one offer is 1 − 0.98^n, which gives 35 applications for 50%, about 87% after 100, and about 114 for 90% (at a 2% chance each).

This assumes every application has the same independent 2% chance, but in real life the odds vary with your skills, the role and how well each application is tailored, so treat the numbers as an illustration rather than a prediction.