How do I approach this problem?
Seven green marbles and three red marbles are randomly placed in Box A and in Box B. Box A holds 4 marbles, while Box B holds 6. What is the probability that Box A has at least 2 green marbles?
There aren't that many cases
We can just list them , box A first, then Box B
GGGG GGGRRR
GGGR GGGGRR
GGRR GGGGGR
GRRR GGGGGG
Since the order does not matter, that is all
of these have 2 or more R's in boxA
so prob = 2/4 = 1/2
It asked for at least 2 G's in A, should read more carefully.
so prob of 2 or more G's in box A is 3/4
To approach this problem, we need to determine the total number of possible outcomes and the number of desired outcomes. Here's how you can solve it step by step:
Step 1: Determine the total number of possible outcomes.
Since there are seven green marbles and three red marbles, the total number of possible outcomes for placing these marbles in Box A can be calculated using combinations. Box A holds four marbles, so the number of combinations can be found using the formula C(7, 4), which is calculated as follows:
C(7, 4) = 7! / (4! * (7 - 4)!) = 7! / (4! * 3!) = (7 * 6 * 5) / (3 * 2 * 1) = 35
Therefore, there are 35 possible combinations of placing marbles in Box A.
Step 2: Determine the number of desired outcomes.
In this case, we want to find the probability that Box A has at least two green marbles. To determine this, we need to calculate the number of combinations where Box A has exactly two, three, or four green marbles.
The number of combinations for two green marbles in Box A can be calculated using C(7, 2), which is equal to:
C(7, 2) = 7! / (2! * (7 - 2)!) = 7! / (2! * 5!) = (7 * 6) / (2 * 1) = 21
The number of combinations for three green marbles in Box A can be calculated using C(7, 3), which is equal to:
C(7, 3) = 7! / (3! * (7 - 3)!) = 7! / (3! * 4!) = (7 * 6 * 5) / (3 * 2 * 1) = 35
The number of combinations for four green marbles in Box A can be calculated using C(7, 4), as we calculated earlier, which is equal to 35.
To find the total number of desired outcomes, we add the combinations for two, three, and four green marbles: 21 + 35 + 35 = 91.
Step 3: Calculate the probability.
Finally, to find the probability of Box A having at least two green marbles, we divide the number of desired outcomes by the total number of possible outcomes:
Probability = Number of desired outcomes / Total number of possible outcomes
Probability = 91 / 35
Therefore, the probability that Box A has at least 2 green marbles is 91/35, or approximately 2.6.