Probability Calculation for Green Balls
This solution explains how to find the probability of randomly selecting two green balls from a box containing balls of different colors.
Total Balls Calculation
- Number of black balls = 2
- Number of green balls = 6
- Number of yellow balls = 4
- Total number of balls = $2 + 6 + 4 = 12$
Total Ways to Pick 2 Balls
We need to find the total number of ways to choose any 2 balls from the 12 available. This is a combination problem, denoted as $C(n, k)$ or $\binom{n}{k}$, where $n$ is the total number of items, and $k$ is the number of items to choose.
Total possible outcomes = $C(12, 2) = \frac{12!}{2!(12-2)!} = \frac{12!}{2!10!} = \frac{12 \times 11}{2 \times 1} = 66$
Favorable Ways to Pick 2 Green Balls
Next, we find the number of ways to choose exactly 2 green balls from the 6 green balls available.
Favorable outcomes = $C(6, 2) = \frac{6!}{2!(6-2)!} = \frac{6!}{2!4!} = \frac{6 \times 5}{2 \times 1} = 15$
Probability Calculation
The probability of an event is the ratio of the number of favorable outcomes to the total number of possible outcomes.
Probability (both balls are green) = $\frac{\text{Favorable outcomes}}{\text{Total possible outcomes}} = \frac{15}{66}$
To simplify the fraction, divide both the numerator and the denominator by their greatest common divisor, which is 3.
Probability (both balls are green) = $\frac{15 \div 3}{66 \div 3} = \frac{5}{22}$
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Important Questions from Probability
Two dice are thrown simultaneously. What is the probability of getting the same number on both the dice?
The probability of being 53 Sundays in year 2020 is-
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When two dice are tossed simultaneously, the probability that the sum of the numbers appearing on both the dice is 8 will be
