Question

1. john is playing a game with two friends. using the spinner pictured, one friend spun a three, and the other friend spun a five. john needs to spin a number higher than both friends in order to win the game, and he wants to calculate his probability of winning. what is the sample space for johns calculation? spinner image with numbers 1,2,3,4,5,6,7,8 options: 2,4,6,8; 1,2,3,4,5; 1,2,3,4,5,6,7,8; 6,7,8

Explanation:

Step1: Understand Sample Space

Sample space is all possible outcomes of a random experiment. Here, the spinner has numbers 1,2,3,4,5,6,7,8. But John needs to spin higher than 3 and 5 (i.e., higher than 5). Wait, no: the friends spun 3 and 5. John needs higher than both, so higher than 5. But sample space for his spin is all possible numbers on the spinner? Wait, no—wait, the question is "What is the sample space for John's calculation?" Wait, no, maybe I misread. Wait, sample space is all possible outcomes of his spin. But let's check the options. Wait, no: the friends spun 3 and 5. John needs to spin higher than both (so higher than 5). But the sample space for his spin is the set of all possible numbers he can spin, but the options: wait, no, maybe the question is about the possible outcomes relevant? Wait, no, sample space is all possible outcomes of the experiment (his spin). The spinner has numbers 1,2,3,4,5,6,7,8. But wait, the options: [6,7,8] is numbers higher than 5 (since 3 and 5, higher than both is >5). Wait, no: 3 and 5, so higher than both means >5. So the possible winning outcomes are 6,7,8, but sample space is all possible outcomes? Wait, no, maybe the question is misphrased. Wait, let's re-express: John's calculation is about his probability of winning, so the sample space for his spin (the experiment) is all possible numbers he can get. But the options: [1,2,3,4,5] is up to 5, [2,4,6,8] is even numbers, [1,2,3,4,5,6,7,8] is all, [6,7,8] is >5. Wait, but sample space is all possible outcomes of the spin, regardless of winning. Wait, but maybe the question is actually asking for the set of possible outcomes that are relevant (winning), but no, sample space is all possible. Wait, no, maybe I made a mistake. Wait, the spinner has 8 sections: 1,2,3,4,5,6,7,8. So the sample space for John's spin (the experiment of spinning the spinner) is {1,2,3,4,5,6,7,8}. But wait, the options include [6,7,8], which are numbers higher than 5 (since 3 and 5, higher than both is >5). Wait, maybe the question is actually asking for the set of possible winning outcomes, but the term "sample space" is the set of all possible outcomes. Wait, no, that can't be. Wait, let's check the options again. The options are:

A. [2,4,6,8] – even numbers

B. [1,2,3,4,5] – numbers ≤5

C. [1,2,3,4,5,6,7,8] – all numbers

D. [6,7,8] – numbers >5

Wait, but sample space is all possible outcomes of the experiment (his spin), so it should be all numbers on the spinner. But that's option C. Wait, but maybe the question is misworded, and it's asking for the set of possible winning outcomes (favorable outcomes), but the term is sample space. Wait, no, sample space is all possible. Wait, let's re-read the question: "What is the sample space for John's calculation?" John is calculating his probability of winning, so the experiment is spinning the spinner, so sample space is all possible outcomes: {1,2,3,4,5,6,7,8}. But wait, the options have [6,7,8] as an option. Wait, maybe the question is actually asking for the set of outcomes that are winning (i.e., higher than both 3 and 5, so >5), but that would be the favorable outcomes, not sample space. But the question says "sample space". Wait, maybe I misread the friends' spins: one spun 3, the other 5. So John needs higher than both, so >5. So the possible winning outcomes are 6,7,8, but sample space is all possible. But the options: [6,7,8] is an option, [1,2,3,4,5,6,7,8] is another. Wait, let's check the spinner: the numbers are 1,2,3,4,5,6,7,8. So the sample space for John's spin is all 8 numbers. But the option [1,2,3,4,5,6,7,8] is option C (third option). Wait, but maybe the question is actually asking for the set of possible outcomes that are relevant to his winning, but that's not sample space. Wait, maybe the question is incorrectly phrased, and it's asking for the set of favorable outcomes (winning), but labeled as sample space. Let's check the options:

- [2,4,6,8]: even numbers, not relevant.

- [1,2,3,4,5]: numbers ≤5, not winning.

- [1,2,3,4,5,6,7,8]: all numbers, sample space.

- [6,7,8]: numbers >5, which are the winning outcomes.

But the question says "sample space for John's calculation". The calculation is about his probability of winning, so the sample space is the set of all possible outcomes of his spin, which is {1,2,3,4,5,6,7,8}. But wait, maybe the question is actually asking for the set of possible outcomes that are considered in his calculation (i.e., the outcomes that matter for winning), but that's the event, not sample space. This is confusing. Wait, let's check the spinner: the numbers are 1,2,3,4,5,6,7,8. So the sample space (all possible outcomes) is {1,2,3,4,5,6,7,8}, which is option C. But wait, the other option [6,7,8] is the set of winning outcomes. Maybe the question has a typo, and it's asking for the favorable outcomes, but labeled as sample space. But according to the definition, sample space is all possible outcomes. So the correct answer should be [1,2,3,4,5,6,7,8]? Wait, no, wait: John's friends spun 3 and 5. He needs to spin higher than both, so higher than 5. So the possible outcomes of his spin are 1-8, but the relevant outcomes for winning are 6-8. But the question is "What is the sample space for John's calculation?" So sample space is all possible outcomes of his spin, which is 1-8. So the option is [1,2,3,4,5,6,7,8], which is the third option. Wait, but let's check the options again:

1. [2,4,6,8] – even numbers

2. [1,2,3,4,5] – numbers ≤5

3. [1,2,3,4,5,6,7,8] – all numbers

4. [6,7,8] – numbers >5

So if sample space is all possible, then option 3. But maybe the question is actually asking for the set of outcomes that are possible for him to win, but that's the event, not sample space. But given the options, and the fact that sample space is all possible, the answer should be [1,2,3,4,5,6,7,8]. Wait, but that seems odd. Wait, no, maybe I misread the problem: "What is the sample space for John's calculation?" John is calculating his probability of winning, so the experiment is his spin, so sample space is all possible results of his spin, which are 1,2,3,4,5,6,7,8. So the correct option is the third one: [1,2,3,4,5,6,7,8]. Wait, but let's confirm: sample space is the set of all possible outcomes of a random experiment. The random experiment here is John spinning the spinner, so the possible outcomes are the numbers on the spinner: 1,2,3,4,5,6,7,8. So the sample space is {1,2,3,4,5,6,7,8}, which matches the third option.

Step2: Evaluate Options

- Option 1: [2,4,6,8] – even numbers, not all possible outcomes. Eliminate.
- Option 2: [1,2,3,4,5] – numbers ≤5, missing 6,7,8. Eliminate.
- Option 3: [1,2,3,4,5,6,7,8] – all numbers on the spinner, matches sample space.
- Option 4: [6,7,8] – numbers >5, these are favorable outcomes (winning), not sample space. Eliminate.

Answer:

[1, 2, 3, 4, 5, 6, 7, 8] (the third option, i.e., the option with [1,2,3,4,5,6,7,8])