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Chapter 3: Problem 20

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 \(p\). If the weather is dry on January 1, show that \(P_{n}\), the probability that it will be dry \(n\) days later, satisfies $$ \begin{aligned} &P_{n}=(2 p-1) P_{n-1}+(1-p) \quad n \geq 1 \\ &P_{0}=1 \end{aligned} $$ Prove that $$ P_{n}=\frac{1}{2}+\frac{1}{2}(2 p-1)^{n} \quad n \geq 0 $$

Short Answer

Expert verified
The probability that it will be dry n days later, \(P_n\), satisfies the recurrence relation \(P_n = (2p - 1)P_{n-1} + (1-p)\) for \(n \geq 1\) and \(P_0 = 1\). Using mathematical induction, we prove the closed form equation \(P_n = \frac{1}{2} + \frac{1}{2}(2p-1)^n\) for \(n \geq 0\).

Step by step solution

01

To derive the recurrence relation, let's first consider the possible outcomes. If the weather is dry n days later, there could be one of two scenarios: 1. The weather was dry (n-1) days prior and the weather remained the same with probability p. 2. The weather was wet (n-1) days prior, and the weather changed to dry with probability (1-p). Therefore, we can calculate the probability P_n as follows: \(P_n = \) (probability of scenario 1) \(+ \) (probability of scenario 2) \(P_n = p P_{n-1} + (1-p)(1-P_{n-1})\) Now, let's simplify this equation: \(P_n = pP_{n-1} - pP_{n-1} + P_{n-1} + p - p^2\) \(P_n = (2p - 1)P_{n-1} + (1-p)\) We have now shown that the recurrence relation for the probability P_n is: \(P_n = (2p - 1)P_{n-1} + (1-p), \quad n \geq 1\), and \(P_0 = 1\) #Step 2: Prove the Closed Form Equation for P_n#

To prove the closed form equation for P_n, we will use mathematical induction. Base Case (n=0): \(P_0 = \frac{1}{2} + \frac{1}{2}(2p-1)^0 = \frac{1}{2} + \frac{1}{2}(1) = 1\) Thus, the closed form equation holds for n=0. Now, let's assume that the closed form equation holds for n=k: \(P_k = \frac{1}{2} + \frac{1}{2}(2p-1)^k\) We want to prove that the equation also holds for n=k+1: \(P_{k+1} = \frac{1}{2} + \frac{1}{2}(2p-1)^{k+1}\) Now, let's use the assumed equation for P_k and the recurrence relation for P_{k+1}: \(P_{k+1} = (2p-1)P_k + (1-p)\) Substitute the assumed equation into the above relation: \(P_{k+1} = (2p-1)\left(\frac{1}{2} + \frac{1}{2}(2p-1)^k\right) + (1-p)\) Now let's simplify the equation: \(P_{k+1} = (2p-1)\left(\frac{1}{2} + \frac{1}{2}(2p-1)^k\right) + (1-p)\) \(P_{k+1} = \frac{(2p-1)}{2} + \frac{(2p-1)^{k+1}}{2} + 1 -p - \frac{(2p-1)}{2}\) \(P_{k+1} = \frac{1}{2} + \frac{1}{2}(2p-1)^{k+1}\) Thus, we have proven by induction that the closed form equation holds for all n: \(P_n = \frac{1}{2} + \frac{1}{2}(2p-1)^n, \quad n \geq 0\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Theory
Probability theory is the branch of mathematics concerned with analyzing random phenomena. A measure of uncertainty is quantified by probability, helping us to predict the likelihood of future events. In the exercise provided, the concept of probability is integral to predicting the weather conditions over a series of days. The probability of the weather remaining the same is denoted by \( p \), and it plays a crucial role in the calculations. We determine the likelihood of future weather being similar to or different from the past based on this probability.
  • The probability of weather remaining the same: \( p \)
  • The probability of weather changing: \( 1 - p \)
Knowing these probabilities allows us to construct models that give us insights into future events, like predicting the weather \( n \) days hence.
Recurrence Relations
Recurrence relations are equations that recursively define a sequence: each term is a function of the preceding terms. In our exercise, the probability \( P_n \) of the weather being dry after \( n \) days is expressed as such a relation. This is represented by the formula:
  • \( P_n = (2p - 1)P_{n-1} + (1-p) \)
This tells us that each term in our sequence, or probability series, is built upon the information from the previous term. By applying this relation repeatedly, we can compute \( P_n \) for any number of days \( n \). Understanding recurrence relations helps break down complex probabilistic rules into simpler, calculable steps.
Mathematical Induction
Mathematical induction is a method of proof used in mathematics to establish that a statement is true for all natural numbers. This method is used to prove the closed-form expression for \( P_n \) in our solution:
  • Base Case: Verify the statement for \( n=0 \)
  • Inductive Step: Assume true for \( n=k \) and prove for \( n=k+1 \)
By proving the base case and then using the assumption to prove the next step, we demonstrate the truth of the statement for all natural numbers. In the exercise, the base case was \( P_0 = 1 \), and we assumed the closed form was true for \( n=k \). Then, we showed it holds for \( n=k+1 \), proving it by induction.
Stochastic Processes
Stochastic processes are collections of random variables representing processes that unfold over time. Weather forecasting, as described in our exercise, is an example of a stochastic process, where the future weather conditions are influenced by probabilistic factors rather than being deterministic.
  • Each day's weather is a random variable dependent on the previous day's state.
  • The process exhibits properties like Markovian dependence, where the future depends only on the present state.
In our exercise, the Markov chain is a type of stochastic process where the probability of the next state (the weather being dry or not) depends only on the current state and not on the history. Stochastic processes like these are foundational in understanding complex systems and predicting outcomes, often used in fields from finance to natural sciences.

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Most popular questions from this chapter

Consider two ums, each containing both white and black balls. The probabilities of drawing white balls from the first and second urns are, respectively, \(p\) and \(p^{\prime}\). Balls are sequentially selected with replacement as follows: With probability \(\alpha\) a ball is initially chosen from the first urn, and with probability \(1-\alpha\) it is chosen from the second urn. The subsequent selections are then made according to the rule that whenever a white ball is drawn (and replaced), the next ball is drawn from the same urn; but when a black ball is drawn, the next ball is taken from the other urn. Let \(\alpha_{n}\) denote the probability that the \(n\)th ball is chosen from the first urn. Show that $$ \alpha_{n+1}=\alpha_{n}\left(p+p^{\prime}-1\right)+1-p^{\prime} \quad n \geq 1 $$ and use this to prove that $$ \alpha_{n}=\frac{1-p^{\prime}}{2-p-p^{\prime}}+\left(\alpha-\frac{1-p^{\prime}}{2-p-p^{\prime}}\right)\left(p+p^{\prime}-1\right)^{n-1} $$ Let \(P_{n}\) denote the probability that the \(n\)th ball selected is white. Find \(P_{n}\). Also compute \(\lim _{n \rightarrow \infty} \alpha_{n}\) and \(\lim _{n \rightarrow \infty} P_{n}\).

Consider 3 urns. Urn \(A\) contains 2 white and 4 red balls; urn \(B\) contains 8 white and 4 red balls; and um \(C\) contains 1 white and 3 red balls. If 1 ball is selected from each urn, what is the probability that the ball chosen from urn \(A\) was white, given that exactly 2 white balls were selected?

Show that $$ \frac{P(H \mid E)}{P(G \mid E)}=\frac{P(H)}{P(G)} \frac{P(E \mid H)}{P(E \mid G)} $$ Suppose that before observing new evidence the hypothesis \(H\) is three times as likely to be true as is the hypothesis \(G\). If the new evidence is twice as likely when \(G\) is true than it is when \(H\) is true, which hypothesis is more likely after the evidence has been observed?

A true-false question is to be posed to a husband and wife team on a quiz show. Both the husband and the wife will, independently, give the correct answer with probability \(p\). Which of the following is a better strategy for this couple? (a) Choose one of them and let that person answer the question; or (b) have them both consider the question and then either give the common answer if they agree or, if they disagree, flip a coin to determine which answer to give?

If two fair dice are rolled, what is the conditional probability that the first one lands on 6 given that the sum of the dice is \(i\) ? Compute for all values of \(i\) between 2 and 12 .
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