a bag of colored marbles holds 71 yellow marbles and 70 white marbles. ten marbles are randomly selected, one at a time, without replacement. what is the probability that more than half are yellow?

To find the probability that more than half of the marbles selected are yellow, we need to determine the number of favorable outcomes and the total number of possible outcomes.

Let's break it down step by step:

1. Determine the total number of possible outcomes:
Since there are 71 yellow marbles and 70 white marbles, the total number of marbles is 71 + 70 = 141.

2. Determine the number of favorable outcomes:
To have more than half yellow marbles, we need to select at least 6 yellow marbles out of the 10 marbles. Let's consider the different possibilities:
- Selecting exactly 6 yellow marbles: We can select 6 yellow marbles out of the 71 in C(71, 6) ways. Similarly, we can select 4 white marbles out of the 70 in C(70, 4) ways. Therefore, the number of favorable outcomes for this case is C(71, 6) * C(70, 4).
- Selecting exactly 7 yellow marbles: We can select 7 yellow marbles out of the 71 in C(71, 7) ways. Similarly, we can select 3 white marbles out of the 70 in C(70, 3) ways. Therefore, the number of favorable outcomes for this case is C(71, 7) * C(70, 3).
- Selecting exactly 8 yellow marbles: We can select 8 yellow marbles out of the 71 in C(71, 8) ways. Similarly, we can select 2 white marbles out of the 70 in C(70, 2) ways. Therefore, the number of favorable outcomes for this case is C(71, 8) * C(70, 2).
- Selecting exactly 9 yellow marbles: We can select 9 yellow marbles out of the 71 in C(71, 9) ways. Similarly, we can select 1 white marble out of the 70 in C(70, 1) ways. Therefore, the number of favorable outcomes for this case is C(71, 9) * C(70, 1).
- Selecting exactly 10 yellow marbles: We can select all 10 yellow marbles out of the 71 in C(71, 10) ways. Therefore, the number of favorable outcomes for this case is C(71, 10).

Now, we need to sum up the favorable outcomes for each case to get the total number of favorable outcomes.

3. Calculate the probability:
The probability is given by the ratio of the number of favorable outcomes to the total number of possible outcomes. Therefore, the probability of selecting more than half yellow marbles is:
(C(71, 6) * C(70, 4) + C(71, 7) * C(70, 3) + C(71, 8) * C(70, 2) + C(71, 9) * C(70, 1) + C(71, 10))/C(141, 10).

By plugging in the values and performing the calculations, you can find the desired probability.