a bowl contains 8 marbles, numbered 1 to 8. Find the probability that in a randomly selected arrangement of 3 marbles, without replacement.
largest # = 5
how to do?
the answer is 6/56
It would help if you proofread your questions before you posted them. What probability are you seeking?
To find the probability of selecting a specific arrangement of 3 marbles from a bowl containing 8 marbles (numbered 1 to 8), without replacement, where the largest numbered marble is 5, you need to determine the number of favorable outcomes and the total number of possible outcomes.
First, let's determine the total number of possible outcomes:
When selecting 3 marbles without replacement from a bowl of 8 marbles, the total number of possible outcomes can be calculated using combinations. In this case, it is represented by "8 choose 3" or written as C(8, 3), which can be calculated as:
C(8, 3) = 8! / (3! * (8 - 3)!) = 8! / (3! * 5!) = (8 * 7 * 6) / (3 * 2 * 1) = 56
Therefore, there are a total of 56 possible outcomes when selecting 3 marbles without replacement from the given bowl.
Next, let's determine the number of favorable outcomes:
Since the largest-numbered marble is 5, we need to consider the possible cases where the largest marble selected does not exceed 5.
We can break it down into two subcases:
Subcase 1: The largest numbered marble is 5
In this subcase, we consider selecting 1, 2, 3, 4, and 5. Since we need to select 3 marbles, we need to choose 2 more marbles from the remaining set of marbles (1, 2, 3, 4). This can be calculated as C(4, 2), which is given by:
C(4, 2) = 4! / (2! * (4 - 2)!) = 4! / (2! * 2!) = (4 * 3) / (2 * 1) = 6
Subcase 2: The largest numbered marble is less than 5
In this subcase, we consider selecting 1, 2, 3, and 4. Since we need to select 3 marbles, we need to choose 3 more marbles from the remaining set of marbles (1, 2, 3, 4). This can be calculated as C(4, 3), which is given by:
C(4, 3) = 4! / (3! * (4 - 3)!) = 4! / (3! * 1!) = 4 / 1 = 4
Therefore, the number of favorable outcomes for both subcases is 6 + 4 = 10.
Finally, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
Probability = Number of favorable outcomes / Total number of possible outcomes
= 10 / 56
≈ 0.1786 (rounded to four decimal places)
Hence, the probability that in a randomly selected arrangement of 3 marbles without replacement, where the largest numbered marble is 5, is approximately 0.1786.