Question

In an urn, there are 5 blue balls and 4 red balls. Four balls are drawn at random from the urn.

Let X be the number of blue balls among the four that are drawn.
Calculate the mode of the distribution of X.

Answer 1

To calculate the mode of the distribution of X (the number of blue balls drawn from the urn), we need to find the most frequently occurring value of X. Using combinatorics, we can calculate the probabilities of different values of X. The mode is the value with the highest probability, which in this case is 2.

Step-by-step explanation:

To calculate the mode of the distribution of X, we need to find the most frequently occurring value of X. X represents the number of blue balls among the four drawn at random from the urn. We can determine the probabilities of different values of X using combinatorics. Let's calculate:

1. The probability of drawing 0 blue balls: P(X=0) = C(4,0) * C(5,4) / C(9,4) = 1 * 5 / 126 = 5/126
2. The probability of drawing 1 blue ball: P(X=1) = C(4,1) * C(5,3) / C(9,4) = 4 * 10 / 126 = 40/126 = 20/63
3. The probability of drawing 2 blue balls: P(X=2) = C(4,2) * C(5,2) / C(9,4) = 6 * 10 / 126 = 60/126 = 10/21
4. The probability of drawing 3 blue balls: P(X=3) = C(4,3) * C(5,1) / C(9,4) = 4 * 5 / 126 = 20/126 = 10/63
5. The probability of drawing 4 blue balls: P(X=4) = C(4,4) * C(5,0) / C(9,4) = 1 * 1 / 126 = 1/126

From the probabilities calculated above, we can see that the mode of the distribution of X is 2, as P(X=2) has the highest probability of 10/21.