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Chapter 4: Problem 39

A ball is drawn from an urn containing 3 white and 3 black balls. After the ball is drawn, it is then replaced and another ball is drawn. This goes on indefinitely. What is the probability that of the first 4 balls drawn, exactly 2 are white?

Short Answer

Expert verified
The probability that exactly 2 out of the first 4 balls drawn are white is \(\frac{3}{8}\).

Step by step solution

01

Determine the probability of success#

In this problem, the probability of success is the probability of drawing a white ball. Since there are 3 white balls and 3 black balls, the total number of balls is 6. The probability of success is then: \(p = \frac{\text{number of white balls}}{\text{total number of balls}} = \frac{3}{6} = \frac{1}{2}\)
02

Determine the number of trials and successful trials#

In this problem, we are considering the first 4 balls drawn. Thus, the number of trials is 4, and we are looking for the probability of exactly 2 successful trials (2 white balls), so \(n=4\) and \(k=2\).
03

Apply the binomial probability formula#

Now we can plug in the values we've found into the binomial probability formula: \(P(X=2) = \binom{4}{2}\left(\frac{1}{2}\right)^2\left(1-\frac{1}{2}\right)^{4-2}\)
04

Calculate the binomial coefficient#

Here, we need to find \(\binom{4}{2}\), which can be calculated using the formula: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\) Plugging in our values, we get: \(\binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4!}{2!2!} = \frac{4\times3\times2\times1}{(2\times1)(2\times1)} = \frac{24}{4} = 6\)
05

Plug in values and calculate the probability#

Now we can plug in the binomial coefficient and the probabilities into the binomial probability formula: \(P(X=2) = 6\left(\frac{1}{2}\right)^2\left(\frac{1}{2}\right)^{4-2} = 6\left(\frac{1}{4}\right)\left(\frac{1}{4}\right) = 6\left(\frac{1}{16}\right) = \frac{6}{16}\)
06

Simplify the result#

Finally, we can simplify the fraction: \(\frac{6}{16} = \frac{3}{8}\)
07

Conclusion#

The probability that exactly 2 out of the first 4 balls drawn are white is \(\frac{3}{8}\).

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Key Concepts

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

Probability Theory Basics
Probability theory is a fundamental principle used to measure the likelihood of an event happening. It is a mathematical framework to quantify uncertainty.
For any given scenario, the probability of a particular outcome is a number between 0 and 1. A probability of 0 means the event will not happen, while 1 means it is certain to happen.
Here's how probabilities are generally calculated:
  • Identify all possible outcomes.
  • Determine the number of ways the successful event can occur.
  • Divide the number of successful outcomes by the total number of possible outcomes.
In our urn problem, probability theory helps determine the likelihood of drawing a specific number of white balls.
Understanding Probability of Success
The probability of success refers to the chance that a chosen event will occur. Here, a 'success' is defined specifically as drawing a white ball.
This is calculated by dividing the number of favorable outcomes (white balls) by the total number of possible outcomes (all balls).
In our example:
  • There are 3 white balls and 3 black balls, totaling 6.
  • The probability of success (drawing a white ball) is given by \( p = \frac{3}{6} = \frac{1}{2} \).
Knowing the probability of success is crucial for further calculations in binomial probability scenarios.
Binomial Coefficient and Its Role
A binomial coefficient is a key element in calculating probabilities in situations where there are multiple trials. It represents the number of ways to choose \( k \) successes in \( n \) trials. Mathematically, it is shown as \( \binom{n}{k} \).
The formula to calculate the binomial coefficient is:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]Where \(!\) denotes factorial, meaning the product of all positive integers up to that number.
In our example:
  • We are drawing 4 balls (\( n = 4 \)) and want exactly 2 of them to be white (\( k = 2 \)).
  • Hence, \( \binom{4}{2} = \frac{4!}{2!2!} = 6 \).
This coefficient is then used in the binomial probability formula.
Trials and Successful Trials in Context
In the context of our problem, a 'trial' refers to the act of drawing a ball from the urn, while a 'successful trial' indicates that a drawn ball is white.
To solve a binomial probability problem effectively:
  • Identify the number of trials, \( n \), which reflects the total rounds or attempts (here, drawing 4 balls).
  • Identify what constitutes a successful trial, and how many such successes \( k \) you are evaluating (drawing 2 white balls).
Understanding these elements allows you to apply and compute the probability using the binomial probability formula:\[P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}\]This guides you to find the probability of obtaining a specific number of 'successful' outcomes in a series of independent trials.

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Most popular questions from this chapter

For a nonnegative integer-valued random variable \(N\), show that $$ E[N]=\sum_{i=1}^{\infty} P\\{N \geq i\\} $$

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