In a box containing 11 red balls, 12 green balls and 5 yellow balls. Two balls are drawn one by one without replacement, what is the probability that
(correct your answers to 4 decimal places.)
both of them are the same color?
both of them are not green?
most likely red since there is more red
really? last I checked, 12 > 11
the first ball can be any color. then there are 27 balls left
P(red,red) = 10/27
P(green,green) = 11/27
P(yellow,yellow) = 4/27
So P(same color) = 25/27
To find the probability in both scenarios, we need to determine the total number of possible outcomes as well as the number of favorable outcomes.
1. Probability that both balls are the same color:
To determine the number of possible outcomes, we need to calculate the total number of ways we can choose two balls out of the total number of balls available. Since we are drawing without replacement, the number of ways can be calculated using combinations.
Total number of ways to choose 2 balls out of 11 red, 12 green, and 5 yellow balls:
= (11+12+5) choose 2
= 28 choose 2
= (28 * 27) / (2 * 1)
= 378
Now, let's determine the number of favorable outcomes. For two balls to have the same color, we can either have both red, both green, or both yellow. So we need to calculate the number of ways we can choose 2 balls out of each color and sum them up.
Number of ways to choose 2 balls out of 11 red balls:
= 11 choose 2
= (11 * 10) / (2 * 1)
= 55
Number of ways to choose 2 balls out of 12 green balls:
= 12 choose 2
= (12 * 11) / (2 * 1)
= 66
Number of ways to choose 2 balls out of 5 yellow balls:
= 5 choose 2
= (5 * 4) / (2 * 1)
= 10
Total favorable outcomes = 55 + 66 + 10 = 131
Now, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
Probability = Total favorable outcomes / Total possible outcomes
= 131 / 378
≈ 0.3460 (rounded to 4 decimal places)
Therefore, the probability that both balls are the same color is approximately 0.3460.
2. Probability that both balls are not green:
To calculate the total possible outcomes, we use the same method as above, considering all 28 balls available.
Total number of ways to choose 2 balls out of the 28 balls:
= 28 choose 2
= (28 * 27) / (2 * 1)
= 378
Now, we need to determine the number of favorable outcomes, which in this case means choosing 2 balls that are not green. There are two scenarios to consider: choosing two red balls or choosing one red ball and one yellow ball.
Number of ways to choose 2 red balls out of 11 red balls:
= 11 choose 2
= (11 * 10) / (2 * 1)
= 55
Number of ways to choose 1 red ball and 1 yellow ball:
= 11 choose 1 * 5 choose 1 (since we have 11 red and 5 yellow balls to choose from)
= 11 * 5
= 55
Total favorable outcomes = 55 + 55 = 110
Probability = Total favorable outcomes / Total possible outcomes
= 110 / 378
≈ 0.2910 (rounded to 4 decimal places)
Therefore, the probability that both balls are not green is approximately 0.2910.