To solve the problem, we need to analyze the two statements regarding the cards marked with consecutive odd numbers from 1 to 200.
### Step 1: Determine the Total Number of Cards
The odd numbers from 1 to 200 are: 1, 3, 5, ..., 199. This forms an arithmetic progression (AP) where:
- First term (a) = 1
- Last term (l) = 199
- Common difference (d) = 2
To find the total number of terms (n) in this AP, we can use the formula for the nth term of an AP:
\[ l = a + (n - 1) \cdot d \]
Substituting the known values:
\[ 199 = 1 + (n - 1) \cdot 2 \]
\[ 199 - 1 = (n - 1) \cdot 2 \]
\[ 198 = (n - 1) \cdot 2 \]
\[ n - 1 = \frac{198}{2} = 99 \]
\[ n = 99 + 1 = 100 \]
**Total number of cards = 100**
### Step 2: Analyze Statement 1
**Statement 1:** Probability that the drawn card is a multiple of 3 is \( \frac{1}{2} \).
To find the number of favorable outcomes (odd multiples of 3):
The odd multiples of 3 between 1 and 199 are: 3, 9, 15, ..., 195. This is also an AP where:
- First term (a) = 3
- Last term (l) = 195
- Common difference (d) = 6
Using the nth term formula:
\[ 195 = 3 + (n - 1) \cdot 6 \]
\[ 195 - 3 = (n - 1) \cdot 6 \]
\[ 192 = (n - 1) \cdot 6 \]
\[ n - 1 = \frac{192}{6} = 32 \]
\[ n = 32 + 1 = 33 \]
**Number of favorable outcomes = 33**
Now, calculate the probability:
\[ P(\text{multiple of 3}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{33}{100} \]
Since \( \frac{33}{100} \neq \frac{1}{2} \), **Statement 1 is false**.
### Step 3: Analyze Statement 2
**Statement 2:** Probability that the drawn card is a perfect square and a multiple of 9 is \( \frac{2}{3} \).
First, find the odd perfect squares between 1 and 199:
The odd perfect squares are: 1, 9, 25, 49, 81, 121, 169.
Next, find which of these are also multiples of 9:
- 9 (which is \( 3^2 \))
- 81 (which is \( 9^2 \))
**Number of favorable outcomes = 2** (9 and 81)
Now, calculate the probability:
\[ P(\text{perfect square and multiple of 9}) = \frac{2}{100} = \frac{1}{50} \]
Since \( \frac{1}{50} \neq \frac{2}{3} \), **Statement 2 is also false**.
### Conclusion
Both statements are false.
