Question

identifying theoretical probability complete each sentence to make a true statement. the theoretical probability of rolling a 2 on a single roll of a number cube is the theoretical probability of landing on blue on a single spin of a spinner with 1 purple, 1 blue, 1 red, and 3 orange section is

Explanation:

First Sentence (Rolling a 2 on a number cube)

Step1: Determine total outcomes

A number cube (die) has 6 faces, so total possible outcomes $n = 6$.

Step2: Determine favorable outcomes

We want to roll a 2, so there's 1 favorable outcome ($m = 1$).

Step3: Calculate theoretical probability

Theoretical probability formula is $P=\frac{m}{n}$. Substituting $m = 1$ and $n = 6$, we get $P=\frac{1}{6}$.

Second Sentence (Spinner with 1 purple, 1 blue, 1 red, and 3 orange sections)

Step1: Determine total sections

Total sections = $1 + 1 + 1 + 3 = 6$? Wait, no—wait, the problem says "1 purple, 1 blue, 1 red, and 3 orange"—wait, 1+1+1+3=6? Wait, no, maybe a typo? Wait, no, let's recheck. Wait, the user's text: "1 purple, 1 blue, 1 red, and 3 orange"—so total sections: 1+1+1+3=6? Wait, but the question is "landing on blue". Favorable outcomes (blue) = 1. Total outcomes = 1+1+1+3=6? Wait, no, maybe I misread. Wait, the original text: "1 purple, 1 blue, 1 red, and 3 orange"—so total sections: 1+1+1+3=6. So favorable (blue) = 1. So probability is $\frac{1}{6}$? Wait, no, maybe the spinner has 1 purple, 1 blue, 1 red, and 3 orange—so total 6 sections. Wait, but maybe the user made a typo. Wait, alternatively, maybe "1 purple, 1 blue, 1 red, and 3 orange"—total 6? Wait, no, 1+1+1+3=6. So landing on blue: 1 favorable, 6 total. So probability $\frac{1}{6}$? Wait, but maybe the original problem has a different total. Wait, maybe the spinner has 1 purple, 1 blue, 1 red, and 3 orange—so total 6. So the theoretical probability of landing on blue is $\frac{1}{6}$? Wait, no, maybe I miscalculated. Wait, let's do it again:

Step1: Calculate total number of sections

Number of purple sections = 1, blue = 1, red = 1, orange = 3. So total sections $n = 1 + 1 + 1 + 3 = 6$.

Step2: Determine favorable outcomes for blue

Number of blue sections $m = 1$.

Step3: Apply probability formula

Theoretical probability $P = \frac{m}{n} = \frac{1}{6}$. Wait, but maybe the user intended a different total? Wait, maybe "1 purple, 1 blue, 1 red, and 3 orange"—no, that's 6. Alternatively, maybe "1 purple, 1 blue, 1 red, and 2 orange"—but the text says 3. So based on the given text, total sections are 6, blue is 1, so probability $\frac{1}{6}$. Wait, but maybe I made a mistake. Alternatively, maybe the spinner has 1 purple, 1 blue, 1 red, and 3 orange—so total 6. So the answer is $\frac{1}{6}$. Wait, but let's confirm.

Wait, the first part: rolling a 2 on a die (number cube) has 6 sides, so probability $\frac{1}{6}$. The second part: spinner with 1 purple, 1 blue, 1 red, 3 orange—total 6 sections. So blue is 1, so probability $\frac{1}{6}$. Wait, but maybe the user's second sentence has a typo, like "3 orange" but maybe it's "2 orange"—but we have to go with the given text. So:

For the spinner:

Step1: Find total sections

$1 + 1 + 1 + 3 = 6$

Step2: Favorable (blue) = 1

Step3: Probability = $\frac{1}{6}$

But maybe the original problem has a different total. Wait, maybe "1 purple, 1 blue, 1 red, and 3 orange"—no, that's 6. So the answers are $\frac{1}{6}$ (for the die) and $\frac{1}{6}$ (for the spinner)? Wait, no, maybe the spinner has 1 purple, 1 blue, 1 red, and 3 orange—total 6, so blue is 1, so $\frac{1}{6}$. Alternatively, maybe the spinner has 1 purple, 1 blue, 1 red, and 3 orange—so total 6. So:

First sentence (die): $\frac{1}{6}$

Second sentence (spinner): $\frac{1}{6}$? Wait, no, maybe the spinner has 1 purple, 1 blue, 1 red, and 3 orange—so total 6, blue is 1, so $\frac{1}{6}$.

But let's proceed with the given text.

Answer:

$\frac{1}{6}$