Question
Samantha is playing games at the fair. In the first game, she gets to throw a softball and can score 0, 2, 4, 6, or 8 points. For the second game, she gets to toss a ring and can score 0, 1, 3, or 5 points. The table below shows all of the possible outcomes for playing each game once. Outcomes 0 1 3 5 0 0, 0 0, 1 0, 3 0, 5 2 2, 0 2, 1 2, 3 2, 5 4 4, 0 4, 1 4, 3 4, 5 6 6, 0 6, 1 6, 3 6, 5 8 8, 0 8, 1 8, 3 8, 5 What is the probability of getting a number greater than 2 in the first game and a number greater than 1 in the second game?
Answers
GPT-4o mini
To find the probability of Samantha scoring more than 2 points in the first game and more than 1 point in the second game, we'll follow these steps:
1. Determine the possible scores for each game.
2. Identify the outcomes that meet the criteria (greater than 2 for the first game, greater than 1 for the second game).
3. Calculate the total number of outcomes and the number of successful outcomes.
### Step 1: Possible Scores
- **First Game (Softball):** Possible scores are 0, 2, 4, 6, 8.
- **Second Game (Ring Toss):** Possible scores are 0, 1, 3, 5.
### Step 2: Successful Outcomes
- For the first game, scores greater than 2 are: **4, 6, 8**
- For the second game, scores greater than 1 are: **3, 5**
Now we need to count the successful pairs of outcomes (from both games):
- For scores **greater than 2** in the first game (4, 6, 8), we have:
- Score 4 from the first game can pair with scores 3 and 5 from the second game:
- (4, 3)
- (4, 5)
- Score 6 from the first game can also pair with scores 3 and 5 from the second game:
- (6, 3)
- (6, 5)
- Score 8 from the first game can also pair with scores 3 and 5 from the second game:
- (8, 3)
- (8, 5)
So the valid pairs are:
1. (4, 3)
2. (4, 5)
3. (6, 3)
4. (6, 5)
5. (8, 3)
6. (8, 5)
This gives us a total of **6 successful outcomes.**
### Step 3: Calculate Total Outcomes
- For the first game, there are 5 possible scores (0, 2, 4, 6, 8).
- For the second game, there are 4 possible scores (0, 1, 3, 5).
- Therefore, the total number of outcomes when playing both games is \(5 \times 4 = 20\).
### Step 4: Probability Calculation
The probability \(P\) of getting a number greater than 2 in the first game and a number greater than 1 in the second game is given by:
\[
P = \frac{\text{Number of successful outcomes}}{\text{Total outcomes}} = \frac{6}{20} = \frac{3}{10}
\]
Thus, the final answer is:
\(\frac{3}{10}\) or 0.3.
1. Determine the possible scores for each game.
2. Identify the outcomes that meet the criteria (greater than 2 for the first game, greater than 1 for the second game).
3. Calculate the total number of outcomes and the number of successful outcomes.
### Step 1: Possible Scores
- **First Game (Softball):** Possible scores are 0, 2, 4, 6, 8.
- **Second Game (Ring Toss):** Possible scores are 0, 1, 3, 5.
### Step 2: Successful Outcomes
- For the first game, scores greater than 2 are: **4, 6, 8**
- For the second game, scores greater than 1 are: **3, 5**
Now we need to count the successful pairs of outcomes (from both games):
- For scores **greater than 2** in the first game (4, 6, 8), we have:
- Score 4 from the first game can pair with scores 3 and 5 from the second game:
- (4, 3)
- (4, 5)
- Score 6 from the first game can also pair with scores 3 and 5 from the second game:
- (6, 3)
- (6, 5)
- Score 8 from the first game can also pair with scores 3 and 5 from the second game:
- (8, 3)
- (8, 5)
So the valid pairs are:
1. (4, 3)
2. (4, 5)
3. (6, 3)
4. (6, 5)
5. (8, 3)
6. (8, 5)
This gives us a total of **6 successful outcomes.**
### Step 3: Calculate Total Outcomes
- For the first game, there are 5 possible scores (0, 2, 4, 6, 8).
- For the second game, there are 4 possible scores (0, 1, 3, 5).
- Therefore, the total number of outcomes when playing both games is \(5 \times 4 = 20\).
### Step 4: Probability Calculation
The probability \(P\) of getting a number greater than 2 in the first game and a number greater than 1 in the second game is given by:
\[
P = \frac{\text{Number of successful outcomes}}{\text{Total outcomes}} = \frac{6}{20} = \frac{3}{10}
\]
Thus, the final answer is:
\(\frac{3}{10}\) or 0.3.