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Chapter 12: Problem 6

Determine the expected counts for each outcome. $$ \begin{array}{lllll} \hline \boldsymbol{n}=\mathbf{7 0 0} & & & & \\ \hline p_{i} & 0.15 & 0.3 & 0.35 & 0.20 \\ \hline{\text {Expected counts }} & & & & \\ \hline \end{array} $$

Short Answer

Expert verified
105, 210, 245, and 140.

Step by step solution

01

- Identify total number of trials

The total number of trials is denoted by \(n\). Here, \(n = 700\).
02

- Identify probabilities for each outcome

The probabilities for each outcome are given as \(p_1 = 0.15, p_2 = 0.3, p_3 = 0.35, p_4 = 0.20\).
03

- Calculate the expected count for each outcome

The formula to calculate the expected count for each outcome is \(E_i = n \times p_i\).
04

- Compute expected count for first outcome

Using the formula: \(E_1 = 700 \times 0.15 = 105\).
05

- Compute expected count for second outcome

Using the formula: \(E_2 = 700 \times 0.3 = 210\).
06

- Compute expected count for third outcome

Using the formula: \(E_3 = 700 \times 0.35 = 245\).
07

- Compute expected count for fourth outcome

Using the formula: \(E_4 = 700 \times 0.20 = 140\).
08

- Compile all expected counts

The expected counts for each outcome are 105, 210, 245, and 140.

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Key Concepts

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

Overview of Probabilities
Understanding probabilities is essential for determining expected counts. Probabilities represent the likelihood that a particular event will occur. In this exercise, the probabilities for four different outcomes were given:
  • \( p_1 = 0.15 \)
  • \( p_2 = 0.3 \)
  • \( p_3 = 0.35 \)
  • \( p_4 = 0.20 \)
All these probabilities add up to 1, which means one of these four outcomes will surely happen in each trial. Probabilities are always between 0 and 1, inclusive.
Number of Trials Explained
The total number of trials, denoted by \(n\), signifies how many times an experiment or event is conducted. In the given exercise, the number of trials is provided as \( n = 700 \). Each trial is an independent instance where one of the four outcomes will occur. The greater the number of trials, the more accurate our expected counts will be, following the Law of Large Numbers. In simple terms, more trials give us a better representation of the probabilities given.
Determining Expected Outcomes
Expected outcomes refer to the predicted frequency of each outcome based on the probabilities and the number of trials. The formula used to compute the expected count for each outcome is:
\[ E_i = n \times p_i \]
Here, \(E_i\) stands for the expected count of the \(i\)-th outcome, \(n\) is the total number of trials, and \(p_i\) is the probability of the \(i\)-th outcome. By substituting the given values in the formula, we found:
  • For the first outcome: \( E_1 = 700 \times 0.15 = 105 \)
  • For the second outcome: \( E_2 = 700 \times 0.3 = 210 \)
  • For the third outcome: \( E_3 = 700 \times 0.35 = 245 \)
  • For the fourth outcome: \( E_4 = 700 \times 0.20 = 140 \)
These calculations show how frequently we can expect each outcome to occur out of 700 trials.
Applying Statistical Formulas
Statistical formulas are crucial for analyzing and interpreting data in experiments. In the context of this exercise, we used the expected count formula:

\[ E_i = n \times p_i \]
This simple yet powerful formula helps to predict the outcome frequencies. Another key aspect is understanding the sum of probabilities. They should total to 1 to ensure all possible outcomes are accounted for. Using these fundamental concepts allows us to effectively interpret and build upon data, leading to accurate insights and informed conclusions. Remember, ensuring accuracy in applying these formulas is key to sound statistical analysis.

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Most popular questions from this chapter

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