There are 5 red and 7 white balls in a bag. Find the probability that at least one ball is red .

Answer

To solve the problem of finding the probability that at least one ball drawn from a bag containing 5 red and 7 white balls is red, we can follow these steps:

### Step 1: Understand the total number of balls
First, we need to determine the total number of balls in the bag. There are 5 red balls and 7 white balls.

**Total number of balls = Number of red balls + Number of white balls**
\[ \text{Total} = 5 + 7 = 12 \]

### Step 2: Define the event of interest
We are interested in finding the probability of drawing at least one red ball. Instead of calculating this directly, it is often easier to calculate the probability of the complementary event (drawing no red balls) and then subtracting that from 1.

### Step 3: Calculate the probability of drawing no red balls
If we want to find the probability of drawing no red balls, we can only draw from the 7 white balls.

**Total outcomes when drawing one ball = Total number of balls = 12**
**Favorable outcomes for drawing no red balls = Number of white balls = 7**

The probability of drawing a white ball (which means no red ball) is:
\[ P(\text{no red}) = \frac{\text{Number of white balls}}{\text{Total number of balls}} = \frac{7}{12} \]

### Step 4: Calculate the probability of drawing at least one red ball
Now, we can find the probability of drawing at least one red ball by subtracting the probability of drawing no red balls from 1:
\[ P(\text{at least one red}) = 1 - P(\text{no red}) = 1 - \frac{7}{12} \]
\[ P(\text{at least one red}) = \frac{12}{12} - \frac{7}{12} = \frac{5}{12} \]

### Final Answer
Thus, the probability that at least one ball drawn is red is:
\[ \frac{5}{12} \]

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