Pick a random number (evenly distributed) between 0 and 1. Continue picking random numbers as long as they keep decreasing; stop picking when you obtain a number that is greater than the previous one you picked. What is the expected number of numbers you pick?

e = 2.718

Step-by-step explanation:

- There're "n!" ways of arranging "n" numbers, supposing that there're n picks, then the first (n−1) picks are in descending order, there are (n−1) ways of choosing the first (n−1) numbers and thus the probability of picking just n numbers is:

                                      (n-1) / n!

- The expected value (E) would be:

                   E = ∑ n*(n-1)/n!

                      = ∑ n*(n-1)/n*(n-1)*(n-2)!

                      = ∑ (n = 2 to infinity) [ 1 / (n-2)! ] = ∑ (n = 0 to infinity) [ 1 / (n)! ]

                      = e   (Maclaurin series approximation)

solution:

the given function is ,

      =

the three roots of this quadratic function is , 0 ,8 and 10.all the roots are real and rational.

if a quadratic equation is of the form ,

sum of the roots = a + b +c= 0+8+10=18=  

product of roots taken two at a time = a b + b c+ c a = 0+80+0=80 =  

product of roots = a b c = 0=  

there are 11 doors in a hall starting from 0 to 11.the entry in the hall is permitted through gate number 0 and exit is allowed through only gate number 8 and 10.so, three roots(0,8,10) of quadratic function is given by entry and exit route of this hall.