Chapter 9.9, Problem 33ES

### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

Chapter
Section

### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# In exercise 23 of Section 9.8, let C k be the event that the gambler has k dollars, wins the next roll of the die, and is eventually runined, let D k be the event that the gamber has k dollars, loses the next roll of the die, and is eventually ruined, and let P k be the probability that the gambler is eventually ruined if has k dollars. Use the probability axioms and the definition of conditional probability to derive the equation P k − 1 = 1 6 P k + 5 6 P k − 2

To determine

To prove:

The equation Pk1=16Pk+56Pk2 using the definition of conditional probability.

Explanation

Given information:

The gambler is betting on 6 when a die is thrown, if the die shows up 6 he wins $1 otherwise he loses$1 each time.

The given Pk is the probability that the gambler is eventually ruined if he has k dollars.

Proof:

The probability that gambler wins =16

And, the probability that gambler loses =56

Pk is the probability that the gambler is eventually ruined if he has k dollars.

Similarly, Pk1 will be the probability that the gambler is eventually ruined if he has k1 dollars

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