Cards marked 1, 3, 5 .... 49 are placed in a box and mixed thoroughly. One card is drawn from the box. Find the probability that the number on the card is :
a composite number

Answer

To find the probability that the number on the card drawn is a composite number, we will follow these steps:

### Step 1: Identify the total number of cards
The cards are marked with odd numbers from 1 to 49. These numbers are:
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49.

To find the total number of cards, we can observe that the sequence of odd numbers from 1 to 49 can be expressed as:
- First term (a) = 1
- Last term (l) = 49
- Common difference (d) = 2

The number of terms (n) in an arithmetic sequence can be calculated using the formula:
\[ n = \frac{l - a}{d} + 1 \]

Substituting the values:
\[ n = \frac{49 - 1}{2} + 1 = \frac{48}{2} + 1 = 24 + 1 = 25 \]

**Total number of cards = 25**

### Step 2: Identify the composite numbers among the cards
Composite numbers are defined as numbers that have more than two distinct positive divisors (not prime). The odd composite numbers from our list are:
- 9 (3 x 3)
- 15 (3 x 5)
- 21 (3 x 7)
- 25 (5 x 5)
- 27 (3 x 9)
- 33 (3 x 11)
- 35 (5 x 7)
- 39 (3 x 13)
- 45 (5 x 9)

Counting these, we find:
- 9, 15, 21, 25, 27, 33, 35, 39, 45

**Total number of composite numbers = 9**

### Step 3: Calculate the probability
The probability (P) of drawing a composite number can be calculated using the formula:
\[ P(\text{composite}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]

Substituting the values we found:
\[ P(\text{composite}) = \frac{9}{25} \]

### Final Answer
The probability that the number on the card is a composite number is:
\[ \frac{9}{25} \]

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