A bag contains 3 black balls 14 green balls and 5 yellow balls.
If two balls are picked at random without replacement , find the probability that balls are of the same colour.
How many black balls must be added so that the probability drawing black balls is a half.
P(black,black) = 3/22 * 2/21
Similarly for the other colors.
Then just add all three values for P(any,same)
Then, you want (3+x)/(22+x) * (2+x)/(21+x) = 1/2
hmm ... I do not get an integer answer...
To find the probability that two balls of the same color are drawn, we need to consider each color separately.
Step 1: Find the probability of drawing two black balls.
The probability of picking a black ball on the first draw is 3/22 (since there are 3 black balls out of 22 total balls).
After removing one black ball, there are 2 black balls left out of 21 remaining balls.
Therefore, the probability of picking a black ball on the second draw is 2/21.
To find the probability of drawing two black balls without replacement, we multiply the individual probabilities:
P(Black, Black) = (3/22) * (2/21) = 6/462 = 1/77.
Step 2: Find the probability of drawing two green balls.
The probability of picking a green ball on the first draw is 14/22.
After removing one green ball, there are 13 green balls left out of 21 remaining balls.
Therefore, the probability of picking a green ball on the second draw is 13/21.
P(Green, Green) = (14/22) * (13/21) = 182/462 = 91/231.
Step 3: Find the probability of drawing two yellow balls.
The probability of picking a yellow ball on the first draw is 5/22.
After removing one yellow ball, there are 4 yellow balls left out of 21 remaining balls.
Therefore, the probability of picking a yellow ball on the second draw is 4/21.
P(Yellow, Yellow) = (5/22) * (4/21) = 20/462 = 10/231.
Step 4: Calculate the total probability of drawing two balls of the same color.
Since we are interested in the probability of drawing two balls of the same color, we need to add the probabilities calculated in steps 1, 2, and 3:
P(Same Color) = P(Black, Black) + P(Green, Green) + P(Yellow, Yellow)
= 1/77 + 91/231 + 10/231
= 97/231.
To find out how many black balls must be added so that the probability of drawing black balls is a half, we set up the following equation:
P(Black) = (number of black balls) / (total number of balls) = 1/2.
Let x be the number of black balls added.
P(Black) = (3 + x) / (22 + x) = 1/2.
Solving for x:
(3 + x) / (22 + x) = 1/2
2(3 + x) = (22 + x)
6 + 2x = 22 + x
2x - x = 22 - 6
x = 16.
Therefore, 16 black balls must be added so that the probability of drawing black balls is a half.