What is the expected number of unique colors among the first ten balls picked, if there are three colors available in the bag?
To find the expected number of unique colors among the first ten balls picked, given that there are three colors available in the bag, we can use the concept of probability.
First, let's calculate the probability of selecting each unique color in a single draw. Since there are three colors available, the probability of selecting a particular color is 1/3.
Now, let's consider selecting the colors in multiple draws. For the first draw, the probability of selecting a unique color is 1 because it is the first ball picked. Therefore, the expected number of unique colors after the first draw is 1.
For the second draw, there are two possibilities. Either we select a new color, with a probability of (2/3), or we select a color that was already picked in the first draw, with a probability of (1/3). If we already have one unique color after the first draw, the expected number of unique colors after the second draw can be calculated as (1 * (2/3)) + (2 * (1/3)) = 4/3.
Similarly, for each subsequent draw, we can calculate the expected number of unique colors based on the previously selected colors and their probabilities.
Using this approach, we can find the expected number of unique colors among the first ten balls picked by calculating the weighted average of the probabilities for each possible outcome. We start with one unique color after the first draw and calculate the expected number of unique colors after each subsequent draw.
Adding up the probabilities for each possible outcome:
(1 * (2/3)) + (4/3 * (1/3)) + (5/3 * (1/3)) + (6/3 * (1/3)) + (7/3 * (1/3)) + (8/3 * (1/3)) + (9/3 * (1/3)) + (10/3 * (1/3)) + (11/3 * (1/3)) + (12/3 * (1/3))
This simplifies to:
(24/9) + (4/9) + (5/9) + (6/9) + (7/9) + (8/9) + (9/9) + (10/9) + (11/9) + (12/9)
Finally, this further simplifies to:
76/9
Therefore, the expected number of unique colors among the first ten balls picked, given that there are three colors available in the bag, is approximately 8.44.
To find the expected number of unique colors among the first ten balls picked, we can make use of the concept of expected value.
Let's denote the variables as follows:
- X: The number of unique colors among the first ten balls picked.
- Xi: A random variable that represents the event that the i-th color is present among the first ten balls picked.
- pi: The probability that the i-th color is present among the first ten balls picked.
Since there are three colors available in the bag, we can assume that each color has an equal probability of being picked, which is 1/3.
Now, let's calculate the probability that the i-th color is present among the first ten balls picked.
For the first color, p1 = 1, as it is always present among the ten balls.
For the second color, p2 = 1 - (1/3)^10, as it is not present in all 10 picks.
For the third color, p3 = 1 - (2/3)^10, as it is not present in any of the 10 picks.
The expected number of unique colors among the first ten balls picked (X) can be found as:
E(X) = Σ(pi * Xi) for i = 1 to 3.
Let's calculate the expected number of unique colors among the first ten balls picked step-by-step:
1. Calculate the probability for each color being present:
p1 = 1
p2 = 1 - (1/3)^10
p3 = 1 - (2/3)^10
2. Calculate the expected number of unique colors:
E(X) = p1 * X1 + p2 * X2 + p3 * X3
Here, X1 = 1 (since the first color is always present),
X2 = 2 (since the second color is present when the first color is not chosen),
X3 = 3 (since the third color is present when neither the first nor the second color is chosen).
3. Substitute the values and calculate:
E(X) = p1 * 1 + p2 * 2 + p3 * 3
Therefore, the expected number of unique colors among the first ten balls picked can be calculated by substituting the probability values and solving the equation.