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
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 boys and girls sit alternatively?
What is the probability that P and Q take the two end positions?
