Nine playing cards are numbered 2 to 10 . A card is selected from them at random .Calculate the probability that the card will be an odd number

Answer

To solve the problem of finding the probability that a randomly selected card from a set of cards numbered 2 to 10 is an odd number, we can follow these steps:

### Step 1: Identify the Sample Space
The sample space consists of all the cards available. The cards are numbered from 2 to 10. Therefore, the sample space \( S \) is:
\[ S = \{2, 3, 4, 5, 6, 7, 8, 9, 10\} \]

### Step 2: Count the Total Number of Outcomes
The total number of cards (outcomes) in the sample space is:
\[ n(S) = 9 \]
This is because there are 9 cards numbered from 2 to 10.

### Step 3: Identify the Favorable Outcomes
Next, we need to identify the odd numbers within the sample space. The odd numbers between 2 and 10 are:
\[ \{3, 5, 7, 9\} \]
Thus, the favorable outcomes \( F \) are:
\[ F = \{3, 5, 7, 9\} \]

### Step 4: Count the Number of Favorable Outcomes
The number of favorable outcomes (odd numbers) is:
\[ n(F) = 4 \]
This is because there are 4 odd numbers in the set.

### Step 5: Calculate the Probability
The probability \( P \) of selecting an odd number is given by the formula:
\[ P(\text{odd number}) = \frac{n(F)}{n(S)} \]
Substituting the values we found:
\[ P(\text{odd number}) = \frac{4}{9} \]

### Final Answer
The probability that the card selected will be an odd number is:
\[ \frac{4}{9} \]

---