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Statistics / A First Course in Probability 9 / Chapter 3 / Problem 22TE

A First Course in Probability | 9th Edition | ISBN: 9780321794772 | Authors: Sheldon Ross

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As a simplified model for weather forecasting, suppose

Chapter 3, Problem 22TE

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Problem 22TE

As a simplified model for weather forecasting, suppose that the weather (either wet or dry) tomorrow will be the same as the weather today with probability p. Show that the weather is dry on January 1, then Pn, the probability that it will be dry n days later, satisfies

Prove that

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QUESTION:

Problem 22TE

As a simplified model for weather forecasting, suppose that the weather (either wet or dry) tomorrow will be the same as the weather today with probability p. Show that the weather is dry on January 1, then Pn, the probability that it will be dry n days later, satisfies

Prove that

ANSWER:

Step 1 of 2

We have to prove that $\ P{\ }_{n}=\frac{1}{2}+\frac{1}{2}(2p-1){\ }^{n};\ n\geq{}0$

Given that p is the probability of the tomorrow’s weather will be same like today’s

And $\ P{\ }_{n}$ is the probability that it will be dry n days later, satisfies the condition

$\ P{\ }_{n}=(2p-1)P{\ }_{n-1}+(1-P);\ n\geq{}1$ …….(1)

And $\ P{\ }_{0}=1$ …….(2)

the induction hypothesis says that for any integer, if $P(K)$ is true then $P(K+1)$ also true. From the first principle of mathematical induction, if the basic step and the inductive steps are proven, then $P(n)$ is true for all natural numbers

Now $P{\ }_{1}=(2p-1)P{\ }_{0}+(1-p)$ [ From equation (1) ]

= $(2p-1)1+(1-p)\ \$ [From Equation (2) ]

= $p$

$=\frac{1}{2}+p-\frac{1}{2}$

$P{\ }_{1}=\frac{1}{2}+\frac{1}{2}(2p-1){\ }^{1}$

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