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Chapter 8: Problem 4

Draw balls from a hat. Suppose there are 40 balls in a hat, of which 10 are red, 10 are blue, 10 are yellow, and 10 are purple. What is the prohahility of getting two blue and two purple balls when drawing 10 balls at random from the hat? Name of program file: 4 balls_from10.py.

Short Answer

Expert verified
The probability of drawing 2 blue and 2 purple balls is approximately 0.1421.

Step by step solution

01

Understanding the Problem

We have a total of 40 balls divided equally among four colors: red, blue, yellow, and purple, with 10 balls each. We need to find the probability of drawing exactly two blue balls and two purple balls when randomly drawing 10 balls from the hat.
02

Use of Combinatorics

The problem involves selecting specific colored balls from a total set of balls, which can be solved using combinations. A combination describes ways to choose items from a group without regard to the order of selection.
03

Calculate Total Ways to Choose 10 Balls

We calculate the total number of ways to choose 10 balls out of the 40 available using the combination formula \( \binom{n}{k} \), where \( n \) is the total number of items to choose from, and \( k \) is the number of items to choose. Here, \( n = 40 \) and \( k = 10 \).\[ \text{Total Ways} = \binom{40}{10} \]
04

Calculate Ways to Choose 2 Blue and 2 Purple Balls

We need to choose 2 blue balls from the available 10 blue balls and 2 purple balls from the available 10 purple balls. The number of ways to choose 2 blue balls is \( \binom{10}{2} \), and the number of ways to choose 2 purple balls is \( \binom{10}{2} \).\[ \text{Ways to choose 2 blue balls} = \binom{10}{2} \]\[ \text{Ways to choose 2 purple balls} = \binom{10}{2} \]
05

Complete the Set with Other Balls

After choosing 4 balls (2 blue and 2 purple), we need to choose 6 more balls from the remaining 30 balls (10 red, 8 blue, 8 yellow, 8 purple). The number of ways to choose these 6 balls is \( \binom{30}{6} \).\[ \text{Ways to choose remaining 6 balls} = \binom{30}{6} \]
06

Calculate Specific Outcome Combination Ways

Calculate the total number of specific ways of choosing 2 blue and 2 purple (which is the product of combinations calculated earlier) plus 6 other balls.\[ \text{Specific Ways} = \binom{10}{2} \times \binom{10}{2} \times \binom{30}{6} \]
07

Calculate Probability of Desired Outcome

Finally, calculate the probability of the favorable outcome by dividing the specific outcome combination ways by the total ways to choose 10 balls.\[ P(2\, \text{blue}\, \text{and}\, 2\, \text{purple}) = \frac{\text{Specific Ways}}{\text{Total Ways}} \]
08

Computation

Compute the values:\[ \binom{40}{10} = 847660528\binom{10}{2} = 45\binom{30}{6} = 593775\]Thus the Specific Ways are:\( 45 \times 45 \times 593775 = 120545625\).Therefore,\[ P(2\, \text{blue}\, \text{and}\, 2\, \text{purple}) = \frac{120545625}{847660528} \approx 0.1421 \]

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Key Concepts

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

Probability
Probability is the branch of mathematics that deals with calculating the likelihood of a given event to occur. In simpler terms, it measures how certain we can be of the occurrence of certain events. Probability ranges between 0 and 1. A probability of 0 indicates an impossible event, while a probability of 1 indicates a certain event. For example, the probability of drawing two blue and two purple balls from a hat containing 40 balls is calculated by determining how many ways that specific combination can happen compared to all possible outcomes. In the given exercise, we are interested in the probability of drawing an exact configuration of balls (two blue and two purple) from a total of 10 drawn balls. This is done by dividing the number of successful outcomes by the total number of possible outcomes, which requires understanding combinations—a key concept in calculating probabilities for such scenarios.
Mathematics Education
Mathematics education involves teaching and learning mathematics comprehensively. It aims to develop problem-solving skills and a deep understanding of mathematical concepts, including combinatorics and probability. To master combinatorics, such as in our ball-drawing problem, students need to grasp the concept of combinations, which are ways of selecting items from a larger pool where order doesn't matter. Mathematics education focuses on providing learners with the tools to systematically apply these principles to solve complex problems, enhancing critical thinking and analytical skills. Effective mathematics education also emphasizes the use of exploration and practice, encouraging students to work through different types of problems, like finding specific probabilities. This approach fosters a more intuitive understanding of how mathematics can be applied to real-world situations.
Problem Solving
Problem solving is a crucial aspect of mathematics that involves identifying solutions to various problems using logical and analytical methodologies. In the context of the given exercise, problem solving starts with a clear understanding of the problem, which involves recognizing the need to calculate probability through combinatorial methods. To solve this particular problem, we break down the task into smaller steps:
  • Identify the type of problem: a combinatorial probability problem.
  • Use combinations to determine the number of successful configurations, such as having specific colors in the drawn balls.
  • Understand how to calculate total possibilities, necessary for determining the probability of the desired outcome.
By adopting a step-by-step approach, problem solving can turn complicated problems into manageable tasks, allowing organized and effective computation, which students can apply in other mathematical contexts.
Combinations
Combinations are a fundamental concept in combinatorics used to calculate how many ways a specific subset can be chosen from a larger set, where the order in which items are chosen does not matter. This concept is key when solving probability problems that involve drawing items without replacement, as seen in our exercise with the colored balls.Combinations are represented by the binomial coefficient \( \binom{n}{k} \), which is calculated using the formula:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]where \(!\) denotes factorial, the product of all positive integers up to that number.In our example, using combinations is essential for determining both the total ways to select any 10 balls from 40 and the specific ways to select 2 blue, 2 purple, and then the remaining 6 balls. By understanding how combinations work, students can effectively determine outcomes in probability problems and gain a strong foundation in combinatorial mathematics.

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Most popular questions from this chapter

Guess beer brands. You are presented \(n\) glasses of beer, each containing a different brand. You are informed that there are \(m \geq n\) possible brands in total, and the names of all brands are given. For each glass, you can pay \(p\) euros to taste the beer, and if you guess the right brand, you get \(q \geq p\) euros back. Suppose you have done this before and experienced that you typically manage to guess the right brand \(T\) times out of 100 , so that your probability of guessing the right brand is \(b=T / 100\). Make a function simulate \((m, n, p, q, b)\) for simulating the beer tasting process. Let the function return the amount of money earned and how many correct guesses \((\leq n)\) you made. Call simulate a large number of times and compute the average earnings and the probability of getting full score in the case \(m=n=4, p=3, q=6\), and \(b=1 / m\) (i.e., four glasses with four brands, completely random guessing, and a payback of twice as much as the cost). How much more can you earn from this game if your ability to guess the right brand is better, say \(b=1 / 2 ?\) Name of program file: simulate_beer_tasting.py.

Estimate the probability in a dice game. Make a program for estimating the probability of getting at least one 6 when throwing \(n\) dice. Read \(n\) and the number of experiments from the command line. (To verify the program, you can compare the estimated probability with the exact result \(11 / 36\) when \(n=2\).) Name of program file: one6_ndice.py.

Probabilities of rolling dice. 1\. You throw a die. What is the probability of getting a \(6 ?\) 2\. You throw a die four times in a row. What is the probability of getting 6 all the times? 3\. Suppose you have thrown the die three times with 6 coming up all times. What is the probability of getting a 6 in the fourth throw? 4\. Suppose you have thrown the die 100 times and experienced a 6 in every throw. What do you think about the probability of getting a 6 in the next throw? First try to solve the questions from a theoretical or common sense point of view. Thereafter, make functions for simulating cases 1,2 , and 3 . Name of program file: rolling_dice.py.

Simulate stock prices. A common mathematical model for the evolution of stock prices can be formulated as a difference equation $$ x_{n}=x_{n-1}+\Delta t \mu x_{n-1}+\sigma x_{n-1} \sqrt{\Delta t} r_{n-1} $$ where \(x_{n}\) is the stock price at time \(t_{n}, \Delta t\) is the time interval between two time levels \(\left(\Delta t=t_{n}-t_{n-1}\right), \mu\) is the growth rate of the stock price, \(\sigma\) is the volatility of the stock price, and \(r_{0}, \ldots, r_{n-1}\) are normally distributed random numbers with mean zero and unit standard deviation. An initial stock price \(x_{0}\) must be prescribed together with the input data \(\mu, \sigma\), and \(\Delta t\). We can make a remark that Equation (8.16) is a Forward Euler discretization of a stochastic differential equation for \(x(t)\) $$ \frac{d x}{d t}=\mu x+\sigma N(t) $$ where \(N(t)\) is a so-called white noise random time series signal. Such equations play a central role in modeling of stock prices. Make \(R\) realizations of \((8.16)\) for \(n=0, \ldots, N\) for \(N=5000\) steps over a time period of \(T=180\) days with a step size \(\Delta t=T / N\). Name of program file: stock_prices.py.

Independent vs. dependent random numbers. Generate a sequence of \(N\) independent random variables with values 0 or 1 and print out this sequence without space between the numbers (i.e., as 001011010110111010\() .\) The next task is to generate random zeros and ones that are dependent. If the last generated number was 0 , the probability of generating a new 0 is \(p\) and a new 1 is \(1-p\). Conversely, if the last generated was 1, the probability of generating a new 1 is \(p\) and a new 0 is \(1-p\). Since the new value depends on the last one, we say the variables are dependent. Implement this algorithm in a function returning an array of \(N\) zeros and ones. Print out this array in the condense format as described above. Choose \(N=80\) and try the probabilities \(p=0.5, p=0.8\) and \(p=0.9\). Can you by visual inspection of the output characterize the differences between sequences of independent and dependent random variables? Name of program file: dependent_random_variables.py.
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