A bag constains white, black and red ball only. A ball is draw at random form the bage. If the probability of getting a white balls is `(3)/(10)` and that of a black ball is `(2)/(5)` then find the probability of getting a red ball. If the bag contains 20 black balls, find the total number of balls in the bag.

Answer

To solve the problem step by step, let's break it down into manageable parts.

### Step 1: Understand the given probabilities
We know:
- Probability of getting a white ball, \( P(W) = \frac{3}{10} \)
- Probability of getting a black ball, \( P(B) = \frac{2}{5} \)

### Step 2: Convert the probability of the black ball to a common denominator
To add the probabilities, we need a common denominator. The denominator for \( P(B) \) is 5, which can be converted to 10:
\[ P(B) = \frac{2}{5} = \frac{2 \times 2}{5 \times 2} = \frac{4}{10} \]

### Step 3: Set up the equation for total probability
The total probability of all events (white, black, and red balls) must equal 1:
\[ P(W) + P(B) + P(R) = 1 \]
Substituting the known values:
\[ \frac{3}{10} + \frac{4}{10} + P(R) = 1 \]

### Step 4: Solve for the probability of the red ball
Combine the probabilities of white and black balls:
\[ \frac{3}{10} + \frac{4}{10} = \frac{7}{10} \]
Now, substitute this back into the equation:
\[ \frac{7}{10} + P(R) = 1 \]
To find \( P(R) \):
\[ P(R) = 1 - \frac{7}{10} = \frac{3}{10} \]

### Step 5: Find the total number of balls in the bag
We know the probability of getting a black ball is given by:
\[ P(B) = \frac{\text{Number of black balls}}{\text{Total number of balls}} \]
Let \( X \) be the total number of balls in the bag. We know there are 20 black balls:
\[ \frac{20}{X} = \frac{2}{5} \]

### Step 6: Cross-multiply to solve for \( X \)
Cross-multiplying gives:
\[ 20 \cdot 5 = 2 \cdot X \]
\[ 100 = 2X \]
Dividing both sides by 2:
\[ X = 50 \]

### Final Answer
The probability of getting a red ball is \( \frac{3}{10} \) and the total number of balls in the bag is 50.

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