Let $$\,{X_1},\,\,{X_2}\,\,$$ and $$\,{X_3}\,$$ be independent and identically distributed random variables with the uniform distribution on $$\left[ {0,1} \right].$$ The probability $$p$$ {$${X_1}$$ is the largest} is __________.

Let $$U$$ and $$V$$ be two independent zero mean Gaussian random variables of variances $${1 \over 4}$$ and $${1 \over 9}$$ respectively. The probability $$\,P\left( {3V \ge 2U} \right)\,\,$$ is

Consider two identically distributed zero - mean random variables $$U$$ and $$V.$$ Let the cumulative distribution functions of $$U$$ and $$2V$$ be $$F(x)$$ and $$G(x)$$ respectively. Then for all values of $$x$$

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