Problem 1263

1. Understanding the Problem:
Tom needs to find 6 specific books out of a total of 30 books. We need to determine the expected number of books he must examine before finding all 6 books.

2. Modeling the Problem:
We can model this problem using the concept of expected value in probability. The books are randomly distributed, and we need to find the expected number of books examined to find all 6 specific books.

3. Using the Coupon Collector's Problem:
This problem is analogous to the Coupon Collector's Problem, where we need to collect all 6 different coupons (books in this case). The expected number of trials to collect all $n$ coupons is given by:
$$E(n) = n \left( \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \right)$$
For our problem, $n = 6$.

4. Calculating the Expected Value:
$$E(6) = 6 \left( \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} \right)$$
We need to compute the harmonic series sum for the first 6 terms:
$$H_6 = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6}$$

5. Summing the Harmonic Series:
$$H_6 = 1 + 0.5 + 0.3333 + 0.25 + 0.2 + 0.1667 \approx 2.45$$

6. Multiplying by the Number of Books:
$$E(6) = 6 \times 2.45 = 14.7$$

7. Conclusion:
The expected number of books Tom must examine before finding all 6 books he needs is approximately 14.7.

The final answer is $\boxed{14.7}$