a bag contains 7 red balls, 4 green balls and 5 yellow balls. what is the probability that 3 balls drawn at random are either green or yellow?
To find the probability of drawing 3 balls that are either green or yellow out of a bag containing 7 red balls, 4 green balls, and 5 yellow balls, follow these steps:
Step 1: Find the total number of balls in the bag.
Total balls = 7 (red) + 4 (green) + 5 (yellow) = 16.
Step 2: Find the total number of ways to choose 3 balls from the bag.
Total ways to choose 3 balls = Choose(16, 3) = 16! / (3! * (16-3)!) = 560.
Step 3: Find the total number of ways to choose 3 green balls from the bag.
Ways to choose 3 green balls = Choose(4, 3) = 4.
Step 4: Find the total number of ways to choose 3 yellow balls from the bag.
Ways to choose 3 yellow balls = Choose(5, 3) = 10.
Step 5: Add the total number of ways to choose 3 green and 3 yellow balls.
Total ways to choose 3 green or yellow balls = Ways to choose 3 green balls + Ways to choose 3 yellow balls = 4 + 10 = 14.
Step 6: Find the probability of choosing 3 green or yellow balls.
Probability = Total ways to choose 3 green or yellow balls / Total ways to choose 3 balls
= 14 / 560
= 1 / 40.
Therefore, the probability of drawing 3 balls at random that are either green or yellow is 1/40.
To find the probability of drawing either green or yellow balls from the bag, we need to first determine the total number of outcomes and the number of favorable outcomes.
Total number of outcomes: In this case, we will draw 3 balls from the bag without replacement. Hence, the total number of outcomes can be calculated using the combination formula: C(n, r) = n! / (r!(n-r)!), where n is the total number of balls (7 red + 4 green + 5 yellow = 16) and r is the number of balls drawn (3). So, C(16, 3) = 16! / (3!(16-3)!) = 16! / (3!13!) = (16 * 15 * 14) / (3 * 2 * 1) = 560.
Number of favorable outcomes: We want to find the probability of drawing either green or yellow balls. There are 4 green and 5 yellow balls in the bag, so the number of favorable outcomes can be calculated using the combination formula: C(4, 3) + C(5, 3) = (4! / (3!(4-3)!)) + (5! / (3!(5-3)!)) = (4! / (3!1!)) + (5! / (3!2!)) = 4 + 10 = 14.
Probability: The probability can be calculated by dividing the number of favorable outcomes by the total number of outcomes: P = (Number of favorable outcomes) / (Total number of outcomes) = 14 / 560 = 0.025 or 2.5%.
Therefore, the probability of drawing 3 balls at random from the bag and getting either green or yellow balls is 2.5%.
If the balls are either green or yellow, it means that we do not want any red balls to be drawn out of the 16 balls.
Assuming the balls are drawn at random AND WITHOUT REPLACEMENT, for the first ball, we have 9 choices out of 16. The second: 8 out of 15, and the third: 7 out of 14.
So P(3 G∪Y)=(9/16)(8/15)(7/14)
If the drawn balls are replaced, then the probability that none of the balls is red will be
P(3 G∪Y with replacement)=(9/16)^3
So the