A fair coin is tossed till a head appears for the first time. The probability that the number of required tosses is odd, is

The box $$1$$ contains chips numbered $$3, 6, 9,$$ $$12$$ and $$15$$. The box $$2$$ contains chips numbered $$6, 11, 16, 21$$ and $$26$$. Two chips, one from each box are drawn at random. The numbers written on these chips are multiplied. The probability for the product to be an even number is ______________.

Consider a Gaussian distributed random variable with zero mean and standard deviation $$\sigma .\,\,\,$$ The value of its cumulative distribution function at the origin will be

$${P_x}\left( X \right) = M{e^{\left( { - 2\left| x \right|} \right)}} + N{e^{\left( { - 3\left| x \right|} \right)}}\,\,$$ is the probability density function for the real random variable $$X,$$ over the entire $$x$$-axis, $$M$$ and $$N$$ are both positive real numbers. The equation relating $$M$$ and $$N$$ is