Answer:
≈ 0.864 or 86.4%
Explanation:
To calculate the probability of selecting at least two new balls out of six randomly chosen balls, we need to consider the different scenarios that satisfy this condition.
First, determine the total number of possible ways to select six balls out of the 20 available. This can be calculated using the combination formula:
C(n, r) = n! / (r!(n-r)!)
Where n is the total number of items and r is the number of items being chosen. In this case, n = 20 (10 new + 10 used) and r = 6.
C(20, 6) = 20! / (6!(20-6)!) = 38,760
Now, calculate the number of ways to select at least two new balls. We can approach this by considering three different cases:
Case 1: Selecting exactly two new balls:
We need to choose two new balls out of the ten available and four used balls out of the remaining ten. This can be calculated as:
C(10, 2) * C(10, 4) = (10! / (2!(10-2)!)) * (10! / (4!(10-4)!)) = 45 * 210 = 9,450
Case 2: Selecting exactly three new balls:
We need to choose three new balls out of the ten available and three used balls out of the remaining ten. This can be calculated as:
C(10, 3) * C(10, 3) = (10! / (3!(10-3)!)) * (10! / (3!(10-3)!)) = 120 * 120 = 14,400
Case 3: Selecting four or more new balls:
In this case, we can either select four new balls or all six new balls. The calculation is as follows:
C(10, 4) * C(10, 2) + C(10, 6) = (10! / (4!(10-4)!)) * (10! / (2!(10-2)!)) + (10! / (6!(10-6)!)) = 210 * 45 + 210 = 9,450 + 210 = 9,660
Now, calculate the total number of ways to select at least two new balls by summing up the results from the three cases:
Total ways to select at least two new balls = Case 1 + Case 2 + Case 3
= 9,450 + 14,400 + 9,660
= 33,510
Finally, calculate the probability by dividing the total number of ways to select at least two new balls by the total number of possible selections:
Probability = Total ways to select at least two new balls / Total possible selections
= 33,510 / 38,760
≈ 0.864 or 86.4%