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There are 50 light bulbs in a room each with its own switch. If a light bulb is on, Dick turns it off and if it is off, he turns it on. Initially all light bulbs are off. After 50 flips and assuming that Dick chooses switches to be flipped randomly, what is the expected number of light bulbs on in the room rounded to the nearest natural number?

To be honest, i tried many times but i can;t seem to arrive at an answer. i am stuck. any help would be appreciated!!!

so i have started with this:

Let E

To be honest, i tried many times but i can;t seem to arrive at an answer. i am stuck. any help would be appreciated!!!

so i have started with this:

Let E

_{i}be the event that bulb i is on after 50 flips. This is the case if switch i is chosen an odd number of times. Compute Pr(E_{i}) and let X_{i}be the indicator random variable for E_{i}. Then E[Xi]=Pr(Ei). Now, let X be the total number of bulbs on after 50 flips. X=∑Xi, so E[X]=∑E[Xi]=n⋅Pr(Ei).
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