Bulb lifespan

The probability that the bulb will burn for more than 800 hours is 0.2. There are 3 light bulbs in the hallway. What is the probability that after 800 hours, at least one will be lit?

Final Answer:

p =  0.488

Step-by-step explanation:

$$\begin{gathered} p_0=0.2 \\ n=3 \\ {p}_1=\left(\genfrac{}{}{0ex}{}{3}{1}\right)\cdot p_0^1\cdot (1-p_0{)}^{3-1}=(\genfrac{}{}{0ex}{}{3}{1})\cdot 0.2^1\cdot (1-0.2{)}^{3-1}=3\cdot 0.2^1\cdot 0.8^2=\frac{48}{125}=0.384 \\ {p}_2=\left(\genfrac{}{}{0ex}{}{3}{2}\right)\cdot p_0^2\cdot (1-p_0{)}^{3-2}=(\genfrac{}{}{0ex}{}{3}{2})\cdot 0.2^2\cdot (1-0.2{)}^{3-2}=3\cdot 0.2^2\cdot 0.8^1=\frac{12}{125}=0.096 \\ {p}_3=\left(\genfrac{}{}{0ex}{}{3}{3}\right)\cdot p_0^3\cdot (1-p_0{)}^{3-3}=(\genfrac{}{}{0ex}{}{3}{3})\cdot 0.2^3\cdot (1-0.2{)}^{3-3}=1\cdot 0.2^3\cdot 0.8^0=\frac{1}{125}=0.008 \\ p=p_1+p_2+p_3=0.384+0.096+0.008=0.488 \end{gathered}$$

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