The probability that a purchased light bulb will work is $0.95$.

How many light bulbs need to be bought so that with a probability of $0.99$ there are at least five working ones among them?

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Let's take 6 light bulbs. The probability that at least 5 of them will be working is the sum of the probabilities that exactly 5 of them will be working and that all 6 will be working, which is $6 \cdot 0.95^{5} \cdot 0.05 + 0.95^{6} = 0.9672$.

Let's take 7 light bulbs. The desired probability is $21 \cdot 0.95^{5} \cdot 0.05^{2} + 7 \cdot 0.95^{6} \cdot 0.05 + 0.95^{7} = 0.9962$.

## Answer

7 light bulbs.