Gambling Probabilities?

Question

You have $2 and you need $8. You can reach your goal by a fair gamble, i.e. at each time you bet your chance of winning is p = 0:5; if you win, you get back your wager and your win (an amount equal to your wager); if you lose, you do not get anything back. Let us consider the following two strategies:

(a) bold strategy: at each round your aim is to come as close as possible to your goal; hence, you stake as much of your current fortune as needed to achieve this.

(1) Determine the probability that you reach your goal of $8; and

(2) Determine how many gambles, on average, it will take until you either reach your goal or lose all your money.

(b) a more careful strategy: as long as your capital is less than $4 you bet $1 at each round. As soon as you have $4 or more you start to bet $2 at each play.

(1) Determine the probability that you reach your goal of $8; and

(2) Determine how many gambles, on average, it will take until you either reach your goal or lose all your money.

Answer 1

(1) bold strategy:

1st gamble: 50% of time bank ($2) lost, 50% won - bank=$4.

2nd gamble: 50% of time bank ($4) lost, 50% attain goal ($8).

Half the games are over in one gamble, half in two = average of 1.5 gambles.

Probability of reaching goal of $8 is .25 (25% of time).

(2) more careful strategy:

Games average 7.5 gambles.

Probability of reaching goal of $8 is .25 (25% of time)

These strategy and all other strategies and systems will have this same outcome. Worth noting is that your betting adversary with $6 is able to survive some loss of capital where you, the player with only $2, cannot. The initial bank ratios $6:$2 are the same as the winning ratios.