3. (1b, 8-11) The probability that a purchased light bulb will work is 0.95. How many light bulbs need to be bought to ensure that with a probability of 0.99, there are at least 5 working ones among them?
# Solution.

Let's take 6 light bulbs. The probability that at least 5 of them will work is the sum of the probabilities that 5 will work and 1 will not, and that all 6 will work, i.e.,

$$
C_{6}^{5} \cdot 0.95^{5} \cdot 0.05 + C_{6}^{6} \cdot 0.95^{6} = 6 \cdot 0.95^{5} \cdot 0.05 + 0.95^{6} = 0.9672
$$

Let's take 7 light bulbs. The desired probability is

$$
\begin{aligned}
& C_{7}^{5} \cdot 0.95^{5} \cdot 0.05^{2} + C_{7}^{6} \cdot 0.95^{6} \cdot 0.05 + C_{7}^{7} \cdot 0.95^{7} = \\
= & 21 \cdot 0.95^{5} \cdot 0.05^{2} + 7 \cdot 0.95^{6} \cdot 0.05 + 0.95^{7} = 0.9962
\end{aligned}
$$

The minimum number of light bulbs required is 7.

Answer: 7.