Two Stockbrokers on the floor of the NEw York Stock Exchange

Two Stockbrokers on the floor of the NEw York Stock Exchange ,Alvin and Bob, are interesed in purchasing shares fron a single company. In a given day, Alvin or Bob buys shares with probability p. Assume that Alvin starts the buying process; when he finishes, Bob is allowed to buy, and so on.

Find the probability that Alvin buys shares on a given day.

Solution

The probability the Alvin or Bob buy it = p

Thus,

Probability that Alvin will buy iy on a given day will be a summ of multiple probabilities

Alvin buys it in first time = p

Alvin buys it on his 2nd turn = (1-p) * (1-p) * p = (1-p)² p

Alvin buys it on 3rd turn = (1-p)*(1-p)*(1-p)*(1-p)*p = (1-p)⁴ p

And so on upto infinity

Thus,

Total probability will be sum of all these probailities upto infinity

= p + (1-p)² p + (1-p)⁴ p + (1-p)⁶ p + (1-p)⁸ p + . . . . .

= p [ 1 + (1-p)² + (1-p)⁴ + (1-p)⁶ + . . . . . ]

Summation of the above series =

= p * [ 1 / 1 - (1-p)² ]

= p / ( 2p - p²)

= 1 / (2 - p) or p = 0

Probability that Alvin will buy the stock = 1 / (2-p) or p =0

Hope this helps.