Analyzing the Probability Problem
The question asks for the probability of selecting two green balls when drawing two balls randomly from a box containing balls of different colors. We need to calculate the number of ways to choose 2 green balls versus the total number of ways to choose any 2 balls.
Calculating Probability of Green Balls
Total Number of Balls
First, find the total number of balls in the box:
Total balls = (Number of black balls) + (Number of green balls) + (Number of yellow balls)
Total balls = 2 + 6 + 4 = 12
Total Possible Outcomes
Calculate the total number of ways to choose 2 balls from the 12 available balls. This uses combinations, denoted as $\binom{n}{k} = \frac{n!}{k!(n-k)!}$:
Total combinations = $\binom{12}{2} = \frac{12!}{2!(12-2)!} = \frac{12!}{2!10!} = \frac{12 \times 11}{2 \times 1} = 66$
Favorable Outcomes (Both Green)
Calculate the number of ways to choose 2 green balls from the 6 green balls available:
Favorable combinations = $\binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6!}{2!4!} = \frac{6 \times 5}{2 \times 1} = 15$
Probability Calculation
The probability of an event is the ratio of favorable outcomes to total possible outcomes:
Probability (both green) = $\frac{\text{Favorable combinations}}{\text{Total combinations}}$
Probability (both green) = $\frac{15}{66}$
Simplifying the Probability
Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3:
Probability (both green) = $\frac{15 \div 3}{66 \div 3} = \frac{5}{22}$
Similar Questions
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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?
