IBM Ponder This #126 · October 2008

Consider a frog starting from 0 and jumping from integer to integer. With probability p he makes a jump of +1, with probability (1-p) a jump of -1. Assume .5 < p < 1. Assume each jump is independent. Let B(n,k) denote the binomial coefficient n choose k.

1. a. How fast will the frog move towards +infinity?

   b. How many times on average will the frog visit each non-negative integer?

2. What is the probability the frog will be at 0 after 2*n jumps?

3. Combine 1b and 2 to find the sum from 0 to infinity of (x**n)*B(2*n,n). (0 < x < .25).

As usual we ask that you only submit your original work.

Solution

Best opened after a real attempt

To be added.