A bag contains 50 balls. Some of them are white, some are blue and some are red. The number of white balls is 11 times the number of blue balls. The number of red balls is less than the number of white balls but more than the number of blue balls.If one ball is taken out of random from the bag, what is the probability that it is red?

Explanation

To solve the problem step by step, we will define variables and use the information given in the question.

### Step 1: Define Variables
Let:
- \( B \) = number of blue balls
- \( W \) = number of white balls
- \( R \) = number of red balls

### Step 2: Set Up Equations
From the problem, we know:
1. The total number of balls is 50:
\[ W + B + R = 50 \]
2. The number of white balls is 11 times the number of blue balls:
\[ W = 11B \]
3. The number of red balls is less than the number of white balls but more than the number of blue balls:
\[ B < R < W \]

### Step 3: Substitute \( W \) in the Total Balls Equation
Substituting \( W = 11B \) into the total balls equation:
\[ 11B + B + R = 50 \]
This simplifies to:
\[ 12B + R = 50 \]
From this, we can express \( R \):
\[ R = 50 - 12B \]

### Step 4: Apply the Conditions for \( R \)
Now, we need to satisfy the condition \( B < R < W \):
1. From \( R > B \):
\[ 50 - 12B > B \implies 50 > 13B \implies B < \frac{50}{13} \approx 3.85 \]
Since \( B \) must be a whole number, \( B \) can be 1, 2, or 3.

2. From \( R < W \):
\[ 50 - 12B < 11B \implies 50 < 23B \implies B > \frac{50}{23} \approx 2.17 \]
Thus, \( B \) must be at least 3.

### Step 5: Determine Possible Values for \( B \)
From the inequalities \( 2.17 < B < 3.85 \), the only integer value for \( B \) is 3.

### Step 6: Calculate \( W \) and \( R \)
Now substituting \( B = 3 \):
- Calculate \( W \):
\[ W = 11B = 11 \times 3 = 33 \]
- Calculate \( R \):
\[ R = 50 - 12B = 50 - 12 \times 3 = 50 - 36 = 14 \]

### Step 7: Verify Conditions
- Check \( B < R < W \):
- \( 3 < 14 < 33 \) (True)

### Step 8: Calculate the Probability of Selecting a Red Ball
The probability \( P \) of selecting a red ball is given by:
\[ P(\text{Red}) = \frac{R}{\text{Total Balls}} = \frac{14}{50} = \frac{7}{25} \]

### Final Answer
Thus, the probability that a randomly selected ball is red is:
\[ \frac{7}{25} \]