Understanding the Probability Problem

The question asks for the probability of drawing two white balls from a bag containing both black and white balls. Probability is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes.

Total Number of Balls and Possible Outcomes

First, let's find the total number of balls in the bag:

• Number of black balls = 5
• Number of white balls = 6
• Total number of balls = 5 + 6 = 11

We are drawing two balls at random from these 11 balls. The total number of ways to choose 2 balls from 11 is given by the combination formula:

$\binom{n}{k} = \frac{n!}{k!(n-k)!}$

Here, $n=11$ (total balls) and $k=2$ (balls drawn). So, the total number of possible outcomes is:

$\text{Total outcomes} = \binom{11}{2} = \frac{11!}{2!(11-2)!} = \frac{11!}{2!9!} = \frac{11 \times 10 \times 9!}{ (2 \times 1) \times 9!} = \frac{11 \times 10}{2} = 11 \times 5 = 55$

There are 55 total ways to draw two balls from the bag.

Favorable Outcomes: Drawing Two White Balls

We want to find the probability that both balls drawn are white. We need to find the number of ways to choose 2 white balls from the 6 white balls available in the bag.

Using the combination formula again, with $n=6$ (white balls) and $k=2$ (white balls drawn):

$\text{Favorable outcomes} = \binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6!}{2!4!} = \frac{6 \times 5 \times 4!}{ (2 \times 1) \times 4!} = \frac{6 \times 5}{2} = 3 \times 5 = 15$

There are 15 ways to draw two white balls from the bag.

Calculating the Probability

The probability of an event is the ratio of the number of favorable outcomes to the total number of possible outcomes.

$\text{Probability (Drawing two white balls)} = \frac{\text{Number of ways to draw two white balls}}{\text{Total number of ways to draw two balls}}$

$\text{Probability} = \frac{15}{55}$

This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 5.

$\text{Probability} = \frac{15 \div 5}{55 \div 5} = \frac{3}{11}$

Summary of Calculation

The probability that the two balls drawn are white is $\frac{3}{11}$.