Consider a sample of size 3 drawn in the following manner:...

Name: Problem 36, Chapter 3: A First Course in Probability
Uploaded: 2022-01-22T14:19:39-08:00
Duration: 4 min 20 s
Channel: Wendi Zhao
Description: Consider a sample of size 3 drawn in the following manner: We start with an urn containing 5 white and 7 red balls. At each stage a ball is drawn and its color is noted. The ball is then returned to the um along with an additional ball of the same color. Find the probability that the sample will contain exactly (a) 0 white balls; (b) 1 white ball; (c) 3 white balls; (d) 2 white balls.

Consider a sample of size 3 drawn in the following manner: We start with an urn containing 5 white and 7 red balls. At each stage a ball is drawn and its color is noted. The ball is then returned to the um along with an additional ball of the same color. Find the probability that the sample will contain exactly (a) 0 white balls; (b) 1 white ball; (c) 3 white balls; (d) 2 white balls.

Consider a sample of size 3 drawn in the following manner: We start with an urn containing 5 white and 7 red balls. At each stage a ball is drawn and its color is noted. The ball is then returned to the um along with an additional ball of the same color. Find the probability that the sample will contain exactly
(a) 0 white balls;
(b) 1 white ball;
(c) 3 white balls;
(d) 2 white balls.

Key Concepts

Polya's Urn Model

This model is a classical probability scheme in which an urn contains balls of different colors, and draws are made sequentially with replacement along with the addition of extra balls of the drawn color. This reinforcement mechanism means that the probability of drawing a particular color changes over time, resulting in a self-reinforcing process where early outcomes influence later draws.

Reinforcement Mechanism

A reinforcement mechanism refers to the process where each draw affects the composition of the urn because an extra ball of the drawn color is added after each draw. This creates a dependency among successive draws, as the chance of drawing a given color increases with each occurrence, altering the probabilities dynamically across the sequence of draws.

Dependent Sampling

Dependent sampling in this context means that the probability of drawing a particular color at any stage is not fixed but depends on the outcomes of previous draws. This contrasts with independent sampling where the probability remains unchanged. Here, each draw's result influences the subsequent composition of the urn, leading to a sequence where probabilities are interlinked.

Beta-Binomial Distribution

The Beta-Binomial distribution arises as the marginal distribution for the number of successes (for example, drawing a white ball) in a fixed number of draws from a Polya's urn model. It captures the extra variability introduced by the reinforcement mechanism, with the parameters of the initial composition of the urn and the number of draws determining the resulting probability of a certain count of outcomes.

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