Problem 5 (10 points) Suppose that each time I play a certain gambling game,IWi

Question

Problem 5 (10 points) Suppose that each time I play a certain gambling game,IWi 20 Doiars with probability p, or I lose 10 Dollars with probability 1 - p. Assume I begin with twenty dollars: as soon as I either (i) obtain at least sixty dollars, or (ii) lose all of my money, I stop playing (a) (5 points) Assume first that p = 1 /3. Find the probability that when I stop playing, I will have at least sixty dollars in my possession. (b) (5 points) Estimate the value of p that corresponds to me finishing with at least sixty dollars with probability 1/2 Hint: You may want to usé a computer for solving Problem 5

Explanation / Answer

Gambling Game

Data's Given :

Solution:

He wants to make the total money in hand as 60 dollar.

Let total no of lose in the game be L

Let total no of win be W

It has to satify the condition : 20dollar + W * 20dollar - L * 10 dollar >= 60 dollar

Simlifying : 2W - L >= 4

=> W >= (4 + L) / 2 eq-1

Let 1-p = q

Now we need to find the probability for each value of L

Case 1: where L = 0, W = 2 , Refer eq-1

P(L=0) = p * p = p2 ( 2 wins and zero Lose)

Case 2: L = 1, W = 3

P(L=1) = q * p3

Case 3 : L=2, W = 3

P(L = 2) = q2 * p3

Case 4 : L =3 , W = 4

P(L = 3) = q3 * p4

Similarly, P(L=4) = q4 * p4

P(L=5) = q5 * p5

Total Probability = sum of all probabilities with different values of L

= p2 + q * p3 + q2 * p3 + q3 * p4 +  q4 * p4 +  q5 * p5 + ..........

= [ p2 + q2 * p3 +  q4 * p4 + ..........] + [ q * p3 + q3 * p4  +  q5 * p5 + ..........] ( Arranging the above equation forms two GP( Geometric Progression ) )

= [ p2 / ( 1 - q2 * p ) ] + [ q * p3 / ( 1 - q2 * p ) ]

=  [ p2 / ( 1 - q2 * p ) ] * [1 + q * p]

(a) p = 1/3 , q = 2/3, Probability = 11/69

(b) p = 1/2, q = 1/2, Probability = 5/14

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