Problem Breakdown:

The problem is about a spinner divided into six equal sectors, numbered 1 to 6.

Part (a): Probability Distribution

To show the probability distribution:

1. Since there are six equal sectors, each outcome (1, 2, 3, 4, 5, and 6) is equally likely.
2. The probability of each number is:

$P(\text{any number}) = \frac{1}{6}.$

We will represent this distribution in:

• A table (listing numbers and their probabilities).
• A graph (showing the probability for each number).

Part (b): Expected Outcome

The expected value $E(X)$ is the average result over many spins, calculated as:

$E(X) = \sum_{i=1}^6 x_i P(x_i),$

where:

• $x_i$ represents each number (1 to 6),
• $P(x_i) = \frac{1}{6}$ is the probability of each number.

Now, I'll calculate both the probability distribution and expected value. matplotlib.pyplot as plt

Data for the probability distribution

numbers = [1, 2, 3, 4, 5, 6] probabilities = [1/6] * 6 # Each number has an equal probability of 1/6

Expected value calculation

expected_value = sum([number * prob for number, prob in zip(numbers, probabilities)])

Create a table for display (numbers and their probabilities)

probability_table = list(zip(numbers, probabilities))

Plot the probability distribution graph

plt.bar(numbers, probabilities, color='skyblue', edgecolor='black') plt.xlabel('Spinner Outcome') plt.ylabel('Probability') plt.title('Probability Distribution of a Single Spin') plt.xticks(numbers) plt.ylim(0, 1) plt.grid(axis='y', linestyle='--', alpha=0.7) plt.show()

Return results

probability_table, expected_value### Solution:

(a) Probability Distribution:

The probability distribution for the spinner's outcomes is shown below:

The bar graph above visually represents the probability distribution. Each outcome has an equal probability of $\frac{1}{6}$, and the graph confirms this uniformity.

(b) Expected Value:

The expected value $E(X)$ is calculated as:

$E(X) = \sum_{i=1}^6 x_i P(x_i) = (1)(\frac{1}{6}) + (2)(\frac{1}{6}) + (3)(\frac{1}{6}) + (4)(\frac{1}{6}) + (5)(\frac{1}{6}) + (6)(\frac{1}{6})$

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

The expected value is $\mathbf{3.5}$.

Interpretation of Expected Value:

The expected value represents the average outcome over a large number of spins. While you can never actually spin a 3.5, it is the mean value you'd expect if you spun the spinner repeatedly.

Would you like more details or further explanation?

Here are 5 related questions for practice:

1. What is the probability of spinning an even number on this spinner?
2. How does the expected value change if the spinner numbers are replaced with 2, 4, 6, 8, 10, and 12?
3. Can the expected value ever be one of the possible outcomes on the spinner? Why or why not?
4. What is the variance and standard deviation of the spinner's outcomes?
5. How would the probability distribution change if the spinner had unequal sectors?

Tip: Expected value is a theoretical average—useful for predicting outcomes in probabilistic systems over time.