A person draws out two balls successively from a bag containing 6 red and 4 white balls . The probability that atleast one of them will be red is

Options

A. `12/90`
B. `30/90`
C. `48/90`
D. `78/90`

Answer

To solve the problem of finding the probability that at least one of the two balls drawn from a bag containing 6 red and 4 white balls is red, we can use the complementary probability approach.

### Step-by-Step Solution:

1. **Identify Total Balls**:
The total number of balls in the bag is:
\[ 6 \text{ (red)} + 4 \text{ (white)} = 10 \text{ (total balls)} \]

**Hint**: Always start by identifying the total number of items in your sample space.

2. **Find Probability of Drawing No Red Balls**:
To find the probability of drawing at least one red ball, it's easier to first calculate the probability of drawing no red balls (i.e., both balls drawn are white).

- The probability of drawing a white ball on the first draw:
\[ P(\text{White first}) = \frac{4}{10} \]

- After drawing one white ball, there are now 3 white balls left and 9 balls total. The probability of drawing a white ball on the second draw:
\[ P(\text{White second | White first}) = \frac{3}{9} \]

- Therefore, the probability of drawing two white balls in a row is:
\[ P(\text{Both White}) = P(\text{White first}) \times P(\text{White second | White first}) = \frac{4}{10} \times \frac{3}{9} = \frac{12}{90} \]

**Hint**: Use conditional probability to find the probability of sequential events.

3. **Calculate Probability of At Least One Red Ball**:
The probability of at least one red ball being drawn is the complement of the probability of drawing no red balls:
\[ P(\text{At least one Red}) = 1 - P(\text{Both White}) = 1 - \frac{12}{90} \]

- Simplifying this gives:
\[ P(\text{At least one Red}) = 1 - \frac{12}{90} = \frac{90 - 12}{90} = \frac{78}{90} \]

**Hint**: Remember that the complement rule states that the probability of an event occurring is equal to 1 minus the probability of it not occurring.

4. **Simplify the Probability**:
We can simplify \(\frac{78}{90}\):
\[ \frac{78}{90} = \frac{39}{45} = \frac{13}{15} \]

**Hint**: Always simplify your fractions to their lowest terms for clarity.

### Final Answer:
The probability that at least one of the two balls drawn is red is:
\[ \frac{13}{15} \]