There are two urns `Aa n dB` . Urn `A` contains 5 red, 3 blue and 2 white balls, urn `B` contains 4 re3d, 3 blue, and 3 white balls. An urn is chosen at random and a ball is drawn. Probability that the ball drawn is red is `9//10` b. `1//2` c. `11//20` d. `9//20`

Options

A. `9//10`
B. `1//2`
C. `11//20`
D. `9//20`

Explanation

To solve the problem, we will calculate the probability of drawing a red ball from either urn A or urn B. Let's break it down step by step.

### Step 1: Define the Events
Let:
- Event A: Choosing urn A
- Event B: Choosing urn B
- Event R: Drawing a red ball

### Step 2: Calculate the Probability of Choosing Each Urn
Since there are two urns and one is chosen at random:
- Probability of choosing urn A, \( P(A) = \frac{1}{2} \)
- Probability of choosing urn B, \( P(B) = \frac{1}{2} \)

### Step 3: Calculate the Probability of Drawing a Red Ball from Each Urn
- Urn A contains 5 red, 3 blue, and 2 white balls. The total number of balls in urn A is \( 5 + 3 + 2 = 10 \).
- Probability of drawing a red ball from urn A, \( P(R|A) = \frac{5}{10} = \frac{1}{2} \)

- Urn B contains 4 red, 3 blue, and 3 white balls. The total number of balls in urn B is \( 4 + 3 + 3 = 10 \).
- Probability of drawing a red ball from urn B, \( P(R|B) = \frac{4}{10} = \frac{2}{5} \)

### Step 4: Use the Law of Total Probability
The total probability of drawing a red ball can be calculated using the law of total probability:
\[ P(R) = P(A) \cdot P(R|A) + P(B) \cdot P(R|B) \]

Substituting the values we calculated:
\[ P(R) = \left(\frac{1}{2} \cdot \frac{1}{2}\right) + \left(\frac{1}{2} \cdot \frac{2}{5}\right) \]

### Step 5: Simplify the Expression
Calculating each term:
- First term: \( \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4} \)
- Second term: \( \frac{1}{2} \cdot \frac{2}{5} = \frac{2}{10} = \frac{1}{5} \)

Now, we need a common denominator to add these fractions. The least common multiple of 4 and 5 is 20:
- Convert \( \frac{1}{4} \) to twentieths: \( \frac{1}{4} = \frac{5}{20} \)
- Convert \( \frac{1}{5} \) to twentieths: \( \frac{1}{5} = \frac{4}{20} \)

Now add them:
\[ P(R) = \frac{5}{20} + \frac{4}{20} = \frac{9}{20} \]

### Final Answer
Thus, the probability that the ball drawn is red is \( \frac{9}{20} \).