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
Three dice are thrown. What is the probability of getting a sum which is a perfect square?
Two distinct natural numbers from 1 to 9 are picked at random. What is the probability that their product has 1 in its unit place?
Two dice are thrown. What is the probability that difference of numbers on them is 2 or 3 ?
Suppose that there is a chance for a newly constructed building to collapse, whether the design is faulty or not. The chance that the design is faulty is 10%. The chance that the building collapses is 95% if the design is faulty, otherwise it is 45%. If it is seen that the building has collapsed, then what is the probability that it is due to faulty design?
What is the probability that all three boys sit together?
