A bag contains 5 red, 3 green, and 6 marbles. If a child grabs three marbles, find P(at least 2 blue marbles).
I assume you mean 6 blue marbles, making 14 in all.
Looking just at the blues,
P(0) = 8/14 * 7/13 * 6/12 = 2/13
P(1) = 6/14 * 8/13 * 7/12 = 2/13
So, P(>=2) = 1-(2/13 + 2/13) = 9/13
To find the probability of getting at least 2 blue marbles, we need to determine the total number of possible outcomes and the number of favorable outcomes.
First, let's calculate the total number of possible outcomes. The child is grabbing 3 marbles from a bag containing 5 red, 3 green, and 6 marbles. Since there are a total of 5 + 3 + 6 = 14 marbles in the bag, the child has 14 choose 3 possible outcomes.
The formula to calculate the number of combinations is given by:
nCr = n! / (r! * (n-r)!)
Using this formula, we can calculate the total number of possible outcomes as:
14C3 = 14! / (3! * (14-3)!)
= 14! / (3! * 11!)
= (14 * 13 * 12) / (3 * 2 * 1)
= 364
Now, let's calculate the number of favorable outcomes (getting at least 2 blue marbles). There are two scenarios where this is possible:
1) Getting exactly 2 blue marbles and 1 of any other color:
The number of ways to choose 2 blue marbles from the 5 available is 5C2.
The number of ways to choose 1 marble of any other color from the remaining 9 marbles is 9C1.
So, the total number of favorable outcomes for this scenario is 5C2 * 9C1 = (5 * 4) * 9 = 180.
2) Getting exactly 3 blue marbles:
The number of ways to choose 3 blue marbles from the 5 available is 5C3.
So, the number of favorable outcomes for this scenario is 5C3 = (5 * 4 * 3) / (3 * 2 * 1) = 10.
Therefore, the total number of favorable outcomes for at least 2 blue marbles is 180 + 10 = 190.
Finally, we can calculate the probability:
P(at least 2 blue marbles) = Number of favorable outcomes / Total number of possible outcomes
= 190 / 364
≈ 0.522
Hence, the probability of grabbing at least 2 blue marbles is approximately 0.522.