Diane has decided to play the following game of chance. She places a $1 bet on each repeated play of the game in which the probability of her winning $1 is 0.3. She has further decided to continue playing the game until she has either accumulated a total of $3 or has lost all her money.
What is the probability that Diane will eventually leave the game a winner if she started with a capital of $2?

A) 0.63
B) 0.27
C) 0.21
D) 0.38

Question

Diane has decided to play the following game of chance. She places a $1 bet on each repeated play of the game in which the probability of her winning $1 is 0.3. She has further decided to continue playing the game until she has either accumulated a total of $3 or has lost all her money. ​
What is the probability that Diane will eventually leave the game a winner if she started with a capital of $2?
​
A) 0.63
B) 0.27
C) 0.21
D) 0.38

Quizplus · Accepted Answer

Let's denote the probability of Diane leaving the game a winner when she starts with $x as P(x). The problem states that Diane wins $1 with a probability of 0.3 and loses $1 with a probability of 0.7 in each game. - When Diane has $0, she cannot play anymore, so P(0) = 0.- When Diane has $3, she has won the game, so P(3) = 1.- For $1 ≤ x ≤ 2, the probability of winning can be expressed in terms of P(x-1) and P(x+1) because, after each game, Diane either has one dollar more or one dollar less. The recursive formula for P(x) is P(x) = 0.3*P(x+1) + 0.7*P(x-1).Given that Diane starts with $2, we need to find P(2). We know P(3) = 1 (since she wins if she has $3) and P(0) = 0 (since she cannot win if she has $0). We can calculate P(1) using the fact that from $1, she either goes to $2 or to $0:P(1) = 0.3*P(2) + 0.7*P(0) = 0.3*P(2) + 0.7*0 = 0.3*P(2).Now, for P(2), she either goes to $3 (and wins) or goes back to $1, so:P(2) = 0.3*P(3) + 0.7*P(1) = 0.3*1 + 0.7*P(1) = 0.3 + 0.7*0.3*P(2).Solving for P(2), we get:P(2) = 0.3 / (1 - 0.7*0.3) = 0.3 / (1 - 0.21) = 0.3 / 0.79 ≈ 0.38.Therefore, the probability that Diane will eventually leave the game a winner if she started with a capital of $2 is approximately 0.38.

Diane Has Decided to Play the Following Game of Chance