An unbiased coin is tossed $$5$$ times. Suppose that a variable $$X$$ is assigned the value $$k$$ when $$k$$ consecutive heads are obtained for $$k=3,4,5,$$ otherwise $$X$$ takes the value $$-1$$. The the expected value of $$X$$, is :
$$k$$ $$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$p(k)$$ $$1/32$$ $$12/32$$ $$11/32$$ $$5/32$$ $$2/32$$ $$1/32$$ k=no. of times head occur simultane $$\displaystyle \sum xp(k)$$ $$=(-1)\times \dfrac{1}{32}+(-1)\times \dfrac{12}{32}+(-1)\times \dfrac{(11)}{32}+3\times \dfrac{5}{32}+4\times \dfrac{2}{32}+5\times \dfrac{1}{32}$$ $$=\dfrac{-1}{32}-\dfrac{12}{32}-\dfrac{11}{32}+\dfrac{15}{32}+\dfrac{8}{32}+\dfrac{5}{32}$$ $$=\dfrac{28-24}{32}$$ $$=\dfrac{4}{32}$$ $$=\dfrac{1}{8}$$
In a workshop, there are five machines and the probability of any one of them to be out of service on day is $$\dfrac{1}{4}$$. If the probability that at most two machines will be out of service on the same day is $$\left(\dfrac{3}{4} \right)^3 k$$, then k is equal to :
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