## 33. In Search of a Job

To find a job after demobilization, soldier Maurice began sending letters to various companies where people of his specialty can be employed. He believes that each of his applications has a one in five chance of being accepted, and he stops sending letters as soon as he finds that he has at least a three in four chance of finding a job.

How many letters will Maurice have to write? (The decimal logarithms of 3, 4, and 5 are approximately 0.477, 0.602, and 0.699.
33. The probability that Maurice will remain unemployed after sending $n$ letters is $(1-1 / 5)^{n}=(4 / 5)^{n}$. Therefore, the probability of finding a job is $1-(4 / 5)^{n}$.

Maurice will stop writing when $n$ becomes such that this probability is not less than $3 / 4$, i.e., when the inequality

$$
(4 / 5)^{n} \leqslant 1 / 4
$$

holds, from which

$$
n \geqslant \frac{\lg 4}{\lg 5-\lg 4}=\frac{0.602}{0.097} \approx 6.2 .
$$

Thus, Maurice will have to write 7 letters.