Probability Calculation: Exactly One White Ball
This solution details the steps to calculate the probability of drawing exactly one white ball when two balls are drawn simultaneously from a bowl containing balls of different colors.
Information Provided
- Number of blue balls = 9
- Number of white balls = 7
- Number of black balls = 5
- Total number of balls = 9 + 7 + 5 = 21
- Number of balls drawn = 2
- Objective: Find the probability of exactly one white ball.
Total Possible Outcomes Calculation
The total number of ways to choose 2 balls from the 21 available balls is determined using combinations ($C(n, k)$).
The formula for combinations is $C(n, k) = \frac{n!}{k!(n-k)!}$.
For this problem, $n=21$ (total balls) and $k=2$ (balls drawn).
Total ways = $C(21, 2) = \frac{21!}{2!(21-2)!} = \frac{21!}{2!19!} = \frac{21 \times 20}{2 \times 1} = 210$.
There are 210 possible ways to draw 2 balls from the bowl.
Favorable Outcomes Calculation
Favorable outcomes involve drawing exactly one white ball and one non-white ball.
- Number of ways to choose 1 white ball from 7 is $C(7, 1) = 7$.
- Number of non-white balls = 9 (blue) + 5 (black) = 14.
- Number of ways to choose 1 non-white ball from 14 is $C(14, 1) = 14$.
The total number of favorable outcomes is the product of these two combinations:
Favorable outcomes = $C(7, 1) \times C(14, 1) = 7 \times 14 = 98$.
Probability Determination
Probability is calculated as the ratio of favorable outcomes to total possible outcomes.
Probability = $\frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{98}{210}$
To simplify the fraction $\frac{98}{210}$, we can divide both the numerator and the denominator by their greatest common divisor, which is 14.
$\frac{98 \div 14}{210 \div 14} = \frac{7}{15}$
The probability of drawing exactly one white ball is $\frac{7}{15}$.
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