An urn contains 10 balls of which 6 are white, 3 are red and 1 is green. Two balls are drawn at random from the urn. Find the probability distribution of number of white balls drawn. Hence state the expected number of white balls one can get in such a draw.
To find the probability distribution of the number of white balls drawn, we need to determine all possible outcomes and their corresponding probabilities.
There are three possible outcomes:
1. Drawing 0 white balls
2. Drawing 1 white ball
3. Drawing 2 white balls
Let's calculate the probabilities for each outcome:
1. Drawing 0 white balls:
The probability of drawing a non-white ball on the first draw is (3 + 1) / 10 = 4 / 10.
After the first non-white ball is drawn, there will be 9 balls left in the urn, of which 5 are white.
The probability of drawing a non-white ball on the second draw is (4 + 1) / 9 = 5 / 9.
Multiplying the probabilities of the two draws gives us the probability of drawing 0 white balls: (4/10) * (5/9) = 2/9.
2. Drawing 1 white ball:
The probability of drawing a white ball on the first draw is 6 / 10.
After the first white ball is drawn, there will be 9 balls left in the urn, of which 5 are white.
The probability of drawing a non-white ball on the second draw is (4 + 1) / 9 = 5 / 9.
Multiplying the probabilities of the two draws gives us the probability of drawing 1 white ball: (6/10) * (5/9) = 1/3.
3. Drawing 2 white balls:
The probability of drawing a white ball on the first draw is 6 / 10.
After the first white ball is drawn, there will be 9 balls left in the urn, of which 5 are white.
The probability of drawing a white ball on the second draw is 5 / 9.
Multiplying the probabilities of the two draws gives us the probability of drawing 2 white balls: (6/10) * (5/9) = 1/3.
Now, let's summarize the probability distribution:
Number of White Balls | Probability
--------------------- | -----------
0 | 2/9
1 | 1/3
2 | 1/3
To find the expected number of white balls, we multiply each possible number of white balls by its corresponding probability and sum the results:
Expected number of white balls = (0 * 2/9) + (1 * 1/3) + (2 * 1/3) = 0 + 1/3 + 2/3 = 1.
Therefore, the expected number of white balls one can get in such a draw is 1.