Question: Members of a Student Council were wanting to get students' opinions on what the theme of the next Spirit Day should be. Complete the two-way table below and use the data to determine the following probabilities.
Let $D = \text{Decades Day}$ and $P = \text{Pajama Day}$
Grade $D$ $P$ Total
9th 163 162 325
10th 155 190 345
11th 176 148 324
12th 198 107 305
Total 692 607 1305
Your answers should be between 0 and 1, inclusive.
$P(10th \text{ or } Decades)$
$P(11th|Pajama)$
$P(Decades|12th)$
Which grade prefers Pajama the most?
9th
10th
11th
12th
1. **State the problem:** We have a two-way table showing the number of students by grade and their preferred Spirit Day theme (Decades or Pajama). We need to find the probabilities:
- $P(10th \text{ or } Decades)$
- $P(11th|Pajama)$
- $P(Decades|12th)$
- Which grade prefers Pajama the most?
2. **Given data:**
| Grade | Decades ($D$) | Pajama ($P$) | Total |
|-------|---------------|--------------|-------|
| 9th | 163 | 162 | 325 |
| 10th | 155 | 190 | 345 |
| 11th | 176 | 148 | 324 |
| 12th | 198 | 107 | 305 |
| Total | 692 | 607 | 1305 |
3. **Calculate $P(10th \text{ or } Decades)$:**
- $P(10th) = \frac{345}{1305}$
- $P(Decades) = \frac{692}{1305}$
- $P(10th \text{ and } Decades) = \frac{155}{1305}$
Use the formula for union:
$$
P(10th \text{ or } Decades) = P(10th) + P(Decades) - P(10th \text{ and } Decades)
$$
Calculate:
$$
P(10th \text{ or } Decades) = \frac{345}{1305} + \frac{692}{1305} - \frac{155}{1305} = \frac{345 + 692 - 155}{1305} = \frac{882}{1305}
$$
Simplify fraction:
$$
\frac{882}{1305} = \frac{\cancel{3}294}{\cancel{3}435} = \frac{294}{435} \approx 0.6759
$$
4. **Calculate $P(11th|Pajama)$:**
Conditional probability formula:
$$
P(11th|Pajama) = \frac{P(11th \text{ and } Pajama)}{P(Pajama)}
$$
Values:
$$
P(11th \text{ and } Pajama) = \frac{148}{1305}, \quad P(Pajama) = \frac{607}{1305}
$$
Calculate:
$$
P(11th|Pajama) = \frac{\frac{148}{1305}}{\frac{607}{1305}} = \frac{148}{1305} \times \frac{1305}{607} = \frac{148}{607} \approx 0.2438
$$
5. **Calculate $P(Decades|12th)$:**
Conditional probability formula:
$$
P(Decades|12th) = \frac{P(Decades \text{ and } 12th)}{P(12th)}
$$
Values:
$$
P(Decades \text{ and } 12th) = \frac{198}{1305}, \quad P(12th) = \frac{305}{1305}
$$
Calculate:
$$
P(Decades|12th) = \frac{\frac{198}{1305}}{\frac{305}{1305}} = \frac{198}{1305} \times \frac{1305}{305} = \frac{198}{305} \approx 0.6492
$$
6. **Determine which grade prefers Pajama the most:**
Calculate $P(Pajama|\text{grade}) = \frac{\text{Pajama count for grade}}{\text{Total students in grade}}$ for each grade:
- 9th: $\frac{162}{325} \approx 0.4985$
- 10th: $\frac{190}{345} \approx 0.5507$
- 11th: $\frac{148}{324} \approx 0.4568$
- 12th: $\frac{107}{305} \approx 0.3508$
The highest is for 10th grade with approximately $0.5507$.
**Final answers:**
- $P(10th \text{ or } Decades) \approx 0.676$
- $P(11th|Pajama) \approx 0.244$
- $P(Decades|12th) \approx 0.649$
- Grade that prefers Pajama the most: 10th