How to gamble if you must
If there was one lesson to be drawn from the discussions in the first part of this book, it was never to gamble when the odds are against you. But, sometimes, we have no choice.
Example 10.1.1Dubrovsky sits down to a night of gambling with his fellow officers. Each time he stakes u roubles, there is a probability p that he will win and receive back 2u roubles (including his stake), and a probability 1 – p that he will lose his stake. At the beginning of the night he has 8000 roubles. If ever he has 256000 roubles he will marry the beautiful Natasha and retire to his estates in the country. Otherwise he will join a monastery. He decides to follow one of two courses of action.
(i) To stake 1000 roubles each time until the issue is decided.
(ii) To stake everything each time until the issue is decided.
Advise him (a) if p = 1/4 and (b) if p = 3/4. What is the probability of a conventional happy ending in each case if he follows your advice?
We know how to calculate the appropriate probabilities if he follows (ii).
Exercise 10.1.2Show that the chances of a happy ending if Dubrovsky follows (ii) are p5. Verify that, if p = 1/4 the probability of a happy ending is roughly 0.001 and, if p = 3/4, roughly 0.237.
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