Question
For a game, a spinner with three equal sections labeled 1 to 3 is spun three times. A player wins if the sum of all three spins is 6 or greater.
Based on the outcome of the first spin, what are all the possible outcomes of the second and third spins that will win? What is the probability of winning based on the outcome of the first spin?
Move possible outcomes and a probability to the spaces to complete the table.
Outcome of First Spin
Possible Outcomes of Second and Third Spins
Probability of Winning Based on the
Outcome of the First Spin
1
2
3
(1, 1)
(2,1)
(3, 1)
1/3
1/9
4/9
(1,2)
(2, 2)
(3, 2)
2/3
7/9
(1,3)
(2,3)
(3,3)
5/9
8/9
2/9
Based on the outcome of the first spin, what are all the possible outcomes of the second and third spins that will win? What is the probability of winning based on the outcome of the first spin?
Move possible outcomes and a probability to the spaces to complete the table.
Outcome of First Spin
Possible Outcomes of Second and Third Spins
Probability of Winning Based on the
Outcome of the First Spin
1
2
3
(1, 1)
(2,1)
(3, 1)
1/3
1/9
4/9
(1,2)
(2, 2)
(3, 2)
2/3
7/9
(1,3)
(2,3)
(3,3)
5/9
8/9
2/9
Answers
GPT-4o mini
To analyze the outcomes of the spins, we begin with the first spin outcome and then determine the possible outcomes for the second and third spins that will yield a total sum of 6 or greater.
Firstly, let's denote the results:
- Let \( x_1 \) be the outcome of the first spin.
- Let \( x_2 \) and \( x_3 \) be the outcomes of the second and third spins, respectively.
The sum requirement is:
\[ x_1 + x_2 + x_3 \geq 6 \]
### Case 1: First Spin = 1
1. \( x_1 = 1 \)
2. We need \( 1 + x_2 + x_3 \geq 6 \) or \( x_2 + x_3 \geq 5 \).
Possible pairs \((x_2, x_3)\) that satisfy this:
- (3, 3) → Total = 7
- (2, 3) → Total = 6
- (3, 2) → Total = 6
The possible outcomes are: **(2, 2), (2, 3), (3, 2), (3, 3)**
**Probability Calculation**:
- Total outcomes for (x_2, x_3) are \( 3 \times 3 = 9 \) (each can be 1, 2, or 3).
- Winning outcomes are 4: (2, 2), (2, 3), (3, 2), (3, 3).
- Thus, the probability is \( \frac{4}{9} \).
### Case 2: First Spin = 2
1. \( x_1 = 2 \)
2. We need \( 2 + x_2 + x_3 \geq 6 \) or \( x_2 + x_3 \geq 4 \).
Possible pairs \((x_2, x_3)\) that satisfy this:
- (1, 3) → Total = 6
- (2, 2) → Total = 6
- (3, 1) → Total = 6
- (2, 3) → Total = 7
- (3, 2) → Total = 7
- (3, 3) → Total = 8
The possible outcomes are: **(1, 3), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)**
**Probability Calculation**:
- Winning outcomes are 6: (1, 3), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3).
- Thus, the probability is \( \frac{6}{9} = \frac{2}{3} \).
### Case 3: First Spin = 3
1. \( x_1 = 3 \)
2. We need \( 3 + x_2 + x_3 \geq 6 \) or \( x_2 + x_3 \geq 3 \).
Possible pairs \((x_2, x_3)\) that satisfy this:
- All combinations are valid since the minimum is (1,1) → Total = 5.
All outcomes are: **(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)**
**Probability Calculation**:
- Total outcomes = 9 (3 choices for x_2, 3 choices for x_3).
- All 9 outcomes win.
- Thus, the probability is \( \frac{9}{9} = 1 \).
### Summary Table
Here’s the completed table with the possible outcomes and probabilities:
| Outcome of First Spin | Possible Outcomes of Second and Third Spins | Probability of Winning Based on the Outcome of the First Spin |
|-----------------------|--------------------------------------------------|----------------------------------------------------------------|
| 1 | (2, 2), (2, 3), (3, 2), (3, 3) | 4/9 |
| 2 | (1, 3), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3) | 2/3 |
| 3 | (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3) | 1 |
This captures all necessary outcomes and probabilities based on the first spin outcomes.
Firstly, let's denote the results:
- Let \( x_1 \) be the outcome of the first spin.
- Let \( x_2 \) and \( x_3 \) be the outcomes of the second and third spins, respectively.
The sum requirement is:
\[ x_1 + x_2 + x_3 \geq 6 \]
### Case 1: First Spin = 1
1. \( x_1 = 1 \)
2. We need \( 1 + x_2 + x_3 \geq 6 \) or \( x_2 + x_3 \geq 5 \).
Possible pairs \((x_2, x_3)\) that satisfy this:
- (3, 3) → Total = 7
- (2, 3) → Total = 6
- (3, 2) → Total = 6
The possible outcomes are: **(2, 2), (2, 3), (3, 2), (3, 3)**
**Probability Calculation**:
- Total outcomes for (x_2, x_3) are \( 3 \times 3 = 9 \) (each can be 1, 2, or 3).
- Winning outcomes are 4: (2, 2), (2, 3), (3, 2), (3, 3).
- Thus, the probability is \( \frac{4}{9} \).
### Case 2: First Spin = 2
1. \( x_1 = 2 \)
2. We need \( 2 + x_2 + x_3 \geq 6 \) or \( x_2 + x_3 \geq 4 \).
Possible pairs \((x_2, x_3)\) that satisfy this:
- (1, 3) → Total = 6
- (2, 2) → Total = 6
- (3, 1) → Total = 6
- (2, 3) → Total = 7
- (3, 2) → Total = 7
- (3, 3) → Total = 8
The possible outcomes are: **(1, 3), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)**
**Probability Calculation**:
- Winning outcomes are 6: (1, 3), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3).
- Thus, the probability is \( \frac{6}{9} = \frac{2}{3} \).
### Case 3: First Spin = 3
1. \( x_1 = 3 \)
2. We need \( 3 + x_2 + x_3 \geq 6 \) or \( x_2 + x_3 \geq 3 \).
Possible pairs \((x_2, x_3)\) that satisfy this:
- All combinations are valid since the minimum is (1,1) → Total = 5.
All outcomes are: **(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)**
**Probability Calculation**:
- Total outcomes = 9 (3 choices for x_2, 3 choices for x_3).
- All 9 outcomes win.
- Thus, the probability is \( \frac{9}{9} = 1 \).
### Summary Table
Here’s the completed table with the possible outcomes and probabilities:
| Outcome of First Spin | Possible Outcomes of Second and Third Spins | Probability of Winning Based on the Outcome of the First Spin |
|-----------------------|--------------------------------------------------|----------------------------------------------------------------|
| 1 | (2, 2), (2, 3), (3, 2), (3, 3) | 4/9 |
| 2 | (1, 3), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3) | 2/3 |
| 3 | (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3) | 1 |
This captures all necessary outcomes and probabilities based on the first spin outcomes.