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 :
divisible by 3

Answer

To solve the problem of finding the probability that a card drawn from a box marked with odd numbers from 1 to 49 is divisible by 3, we can follow these steps:

### Step 1: Identify the total odd numbers from 1 to 49.
The odd numbers from 1 to 49 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.

### Step 2: Count the total number of odd numbers.
To count the total odd numbers, we can observe that they form an arithmetic sequence where the first term \(a = 1\) and the common difference \(d = 2\). The last term \(l = 49\).

The number of terms \(n\) in an arithmetic sequence can be calculated using the formula:
\[ n = \frac{l - a}{d} + 1 \]
Substituting the values:
\[ n = \frac{49 - 1}{2} + 1 = \frac{48}{2} + 1 = 24 + 1 = 25 \]

So, there are 25 odd numbers in total.

### Step 3: Identify the odd numbers that are divisible by 3.
Now, we need to find the odd numbers among these that are divisible by 3. The odd numbers divisible by 3 from the list are:
3, 9, 15, 21, 27, 33, 39, 45.

### Step 4: Count the favorable outcomes.
The favorable outcomes (odd numbers divisible by 3) are:
3, 9, 15, 21, 27, 33, 39, 45.
Counting these gives us a total of 8 favorable outcomes.

### Step 5: Calculate the probability.
The probability \(P\) of drawing a card that is divisible by 3 is given by the formula:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]
Substituting the values:
\[ P = \frac{8}{25} \]

Thus, the probability that the number on the card drawn is divisible by 3 is \(\frac{8}{25}\).

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