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Chapter 3: Problem 16

The probability that a visitor to a Web site provides contact data for additional information is \(0.01 .\) Assume that 1000 visitors to the site behave independently. Determine the following probabilities: a. No visitor provides contact data. b. Exactly 10 visitors provide contact data. c. More than 3 visitors provide contact data.

Short Answer

Expert verified
a. \(0.000043\); b. \(0.125\); c. \(0.761\).

Step by step solution

01

Define the Random Variable

Let \( X \) be the random variable representing the number of visitors providing contact data out of 1000 visitors. \( X \) follows a binomial distribution with parameters \( n = 1000 \) and \( p = 0.01 \).
02

Calculate the Probability for No Visitor

We calculate \( P(X = 0) \) using the binomial probability formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] For \( k = 0 \), this becomes: \[ P(X = 0) = \binom{1000}{0} (0.01)^0 (0.99)^{1000} = (0.99)^{1000} \] Calculating this gives approximately \( P(X = 0) \approx 0.000043 \).
03

Calculate the Probability for Exactly 10 Visitors

For \( k = 10 \), use the binomial formula: \[ P(X = 10) = \binom{1000}{10} (0.01)^{10} (0.99)^{990} \] Calculation of the combination and the expression gives \( P(X = 10) \approx 0.125 \).
04

Calculate the Probability for More Than 3 Visitors

To find \( P(X > 3) \), use the complement rule: \( P(X > 3) = 1 - P(X \leq 3) \), where: \[ P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) \] Calculate each using the binomial formula, then sum them up and subtract from 1. Calculation gives \( P(X > 3) \approx 0.761 \).

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Key Concepts

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

Probability Calculation
Probability calculation is an essential concept in statistics that measures the likelihood of a given event happening. In the context of binomial distribution, it becomes crucial to determine the probability of observing a specific number of successes in a set number of trials.
The exercise presented revolves around a web visitor providing contact data, where the probability of a single visitor doing so is 0.01. Given 1000 visitors, we aim to calculate various probability scenarios, such as no visitor providing contact data, exactly 10 doing so, and more than three.
  • First, the probability of no visitor providing contact data is calculated using the expression for probability with no successes in 1000 trials, which simplifies to directly evaluating \(0.99^{1000}\).
  • For exactly 10 visitors providing contact data, the probability is calculated using the binomial probability formula, which combines the choice of 10 from 1000 (indicated by combinations) with the specifics of the event's probability.
  • Then, the probability of more than 3 visitors is calculated by finding the complement鈥攁 critical strategy in such calculations, which involves summing the probabilities from zero to three and subtracting from one.
These tools provide powerful ways to tackle problems where events are repetitive and independent.
Random Variable
In probability and statistics, a random variable is a variable that takes on different values based on the outcomes of a random phenomenon. It can be thought of as a numerical description of an event.
For the problem at hand, we define the random variable \(X\) as the number of visitors who provide contact information out of the 1000 visitors. Here, the random variable follows a binomial distribution, which is specifically useful for scenarios with two possible outcomes: success or failure.
  • The value of \(X\) can range from 0 to 1000, where each number represents a different potential outcome (e.g., "0" means no visitors provide contact info, whereas "1000" indicates all do).
  • Determining the binomial parameters鈥攏amely, the number of trials \(n = 1000\) and success probability \(p = 0.01\)鈥攑rovides us with all necessary information to perform subsequent probability calculations.
  • This approach assumes each visitor behaves independently, enabling clear probability calculations.
Understanding random variables is foundational for any probability analysis and supports the abstraction needed for more complex stochastic modeling.
Complement Rule
The complement rule is a powerful tool in probability that simplifies the computation of the probability of an event's occurrence by considering its opposite.
This rule states that the probability of an event occurring is equal to one minus the probability of its complement (the event not occurring). This becomes particularly useful in situations where direct computation is challenging.
  • In this exercise, to determine the probability of more than 3 visitors providing contact data, it's easier to calculate the probability of 3 or fewer providing it and subtract that result from 1.
  • The complement concept is grounded in the certainty that between a given event and its complement, one must occur. Therefore, the sum of their probabilities is always 1.
  • Applying this rule streamlines calculations by minimizing the number of direct probability evaluations needed.
The complement rule exemplifies simplicity in probability, making it a favorite tool for statisticians in handling binary-outcome processes.
Parameter Estimation
Parameter estimation involves identifying the values of parameters for a specific statistical population distribution based on observational data. With binomial distribution, these include the number of trials \(n\) and the probability of success on each trial \(p\).
In the problem given, we observe that the parameters are: 1000 total observations (visitors) and a probability of success (contact data provision) of 0.01.
  • These parameters define the characteristics of our binomial distribution and allow us to compute the exact probabilities of particular outcomes.
  • Parameter estimation demands an assumption of independence among observations, which simplifies modeling as every event remains consistent with others in terms of probability.
  • Accurate parameter estimation is critical because it forms the basis upon which predictive insights and conclusions are drawn in any probability distribution scenario.
With correct parameter values, prediction and interpretation remain broadly applicable, enabling effective decision-making from statistical models.

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