To address the problem, we'll first analyze the probability distribution for the spinner and then calculate the expected outcome. Let's break it into the requested parts:

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Part (a): Probability Distribution

1. Table Representation

The spinner has six equal sectors, numbered 1 to 6, meaning the probability of landing on each number is equal. The probability for each sector is:

$P(\text{sector}) = \frac{1}{6}$

Outcome (X)	1	2	3	4	5	6
Probability P(X)	$\frac{1}{6}$	$\frac{1}{6}$	$\frac{1}{6}$	$\frac{1}{6}$	$\frac{1}{6}$	$\frac{1}{6}$

2. Graph Representation

The graph will be a bar graph with each outcome ($1, 2, 3, 4, 5, 6$) on the x-axis and their probabilities ($\frac{1}{6}$) on the y-axis.

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Part (b): Expected Outcome

The expected value $E(X)$ is the weighted average of all possible outcomes. For a discrete probability distribution:

$E(X) = \sum (X \cdot P(X))$

Using the given probabilities and outcomes: $E(X) = 1 \cdot \frac{1}{6} + 2 \cdot \frac{1}{6} + 3 \cdot \frac{1}{6} + 4 \cdot \frac{1}{6} + 5 \cdot \frac{1}{6} + 6 \cdot \frac{1}{6}$

$E(X) = \frac{1 + 2 + 3 + 4 + 5 + 6}{6} = \frac{21}{6} = 3.5$

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Interpretation of the Expected Value

The expected value of $3.5$ means that if you spin the spinner many times, the average outcome over those spins would tend to approach $3.5$. This does not mean you can land on $3.5$ in a single spin (since it's not a possible outcome), but it represents the "center" of the probability distribution.

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Graph Representation

Let me generate the graph for you. matplotlib.pyplot as plt

Data for the spinner

outcomes = [1, 2, 3, 4, 5, 6] probabilities = [1/6] * 6

Create the bar graph

plt.figure(figsize=(8, 5)) plt.bar(outcomes, probabilities, color='skyblue', edgecolor='black')

Add labels and title

plt.xlabel("Outcomes (X)", fontsize=12) plt.ylabel("Probability P(X)", fontsize=12) plt.title("Probability Distribution for a Single Spin", fontsize=14) plt.xticks(outcomes) plt.ylim(0, 0.2)

Show grid for better visualization

plt.grid(axis='y', linestyle='--', alpha=0.7)

Show the plot

plt.tight_layout() plt.show()Here is the bar graph showing the probability distribution for the spinner. Each outcome from 1 to 6 has an equal probability of $\frac{1}{6}$.

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Summary:

• Probability distribution: Equal probabilities ($\frac{1}{6}$) for outcomes 1 to 6.
• Expected outcome: $3.5$, representing the average result over many spins.

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Would you like more details or have any questions? Here are 5 related questions to explore further:

1. How would the probabilities change if the spinner had unevenly sized sectors?
2. How can we calculate the variance and standard deviation for this spinner?
3. What does the expected value mean in practical applications, such as games?
4. How would the graph and expected value differ for a weighted spinner?
5. How can you simulate multiple spins to verify the expected value experimentally?

Tip: Expected value represents a long-term average but does not guarantee any single result.