A box contains 7 red balls 8 white ball & 3 blue ball if three ball are drawn at random from the box without replacement calculate the probability that
A)two are red and one is blue
B)at least two are white
C)one of each color is drawn
A) The probability of drawing two red balls and one blue ball can be found by multiplying the probabilities of each event happening in order.
First, the probability of drawing a red ball on the first draw is 7/18 (since there are 7 red balls out of 18 total).
Next, the probability of drawing another red ball on the second draw (since we're drawing without replacement) is 6/17 (since there will now be 6 red balls left out of 17 total balls remaining).
Finally, the probability of drawing a blue ball on the third draw is 3/16 (since there will be 3 blue balls left out of 16 total balls remaining).
Multiplying these probabilities together gives:
(7/18) * (6/17) * (3/16) = 0.024
So the probability of drawing two red balls and one blue ball is 0.024.
B) The probability of drawing at least two white balls can be found by calculating the probability of drawing two white balls and the probability of drawing three white balls, and adding them together.
The probability of drawing two white balls can be found similarly to part A:
First, the probability of drawing a white ball on the first draw is 8/18 (since there are 8 white balls out of 18 total).
Next, the probability of drawing another white ball on the second draw (since we're drawing without replacement) is 7/17 (since there will now be 7 white balls left out of 17 total balls remaining).
Finally, the probability of drawing a non-white ball on the third draw is 10/16 (since there will be 10 non-white balls left out of 16 total balls remaining).
Multiplying these probabilities together gives:
(8/18) * (7/17) * (10/16) = 0.159
So the probability of drawing two white balls is 0.159.
To find the probability of drawing three white balls, we can simply multiply the probabilities of drawing a white ball on all three draws (again, without replacement):
(8/18) * (7/17) * (6/16) = 0.091
So the probability of drawing three white balls is 0.091.
Adding these two probabilities together gives:
0.159 + 0.091 = 0.25
So the probability of drawing at least two white balls is 0.25.
C) The probability of drawing one ball of each color can be found by multiplying the probabilities of drawing one ball of each color in order.
First, the probability of drawing a red ball on the first draw is 7/18 (since there are 7 red balls out of 18 total).
Next, the probability of drawing a white ball on the second draw (since we're drawing without replacement) is 8/17 (since there will be 8 white balls left out of 17 total balls remaining).
Finally, the probability of drawing a blue ball on the third draw is 3/16 (since there will be 3 blue balls left out of 16 total balls remaining).
Multiplying these probabilities together gives:
(7/18) * (8/17) * (3/16) = 0.038
So the probability of drawing one ball of each color is 0.038.
To calculate the probabilities, we need to find the total number of possible outcomes and the number of favorable outcomes for each event. Let's calculate each probability step-by-step:
A) Probability that two are red and one is blue:
Step 1: Calculate the total number of balls in the box.
Total number of balls = 7 red balls + 8 white balls + 3 blue balls = 18 balls.
Step 2: Calculate the total number of ways to choose 3 balls out of the 18.
Total number of ways to choose 3 balls out of 18 = combination(18, 3) = 18! / (3!(18-3)!) = 816.
Step 3: Calculate the number of favorable outcomes where two are red and one is blue.
Number of ways to choose 2 red balls out of 7 * Number of ways to choose 1 blue ball out of 3 = combination(7, 2) * combination(3, 1) = (7! / (2!(7-2)!)) * (3! / (1!(3-1)!)) = 21 * 3 = 63.
Step 4: Calculate the probability.
Probability = favorable outcomes / total outcomes = 63 / 816 ≈ 0.0772.
Therefore, the probability that two balls are red and one is blue is approximately 0.0772.
B) Probability that at least two are white:
Step 1: Calculate the total number of ways to choose 3 balls out of the 18 (as calculated before) = 816.
Step 2: Calculate the number of favorable outcomes where at least two are white.
Number of ways to choose 2 white balls out of 8 * Number of ways to choose 1 ball from the remaining options (red and blue) = combination(8, 2) * combination(10, 1) = (8! / (2!(8-2)!)) * (10! / (1!(10-1)!)) = 28 * 10 = 280.
Step 3: Calculate the probability.
Probability = favorable outcomes / total outcomes = 280 / 816 ≈ 0.3431.
Therefore, the probability that at least two balls are white is approximately 0.3431.
C) Probability that one of each color is drawn:
Step 1: Calculate the total number of ways to choose 3 balls out of the 18 (as calculated before) = 816.
Step 2: Calculate the number of favorable outcomes where one of each color is drawn.
Number of ways to choose 1 red ball out of 7 * Number of ways to choose 1 white ball out of 8 * Number of ways to choose 1 blue ball out of 3 = combination(7, 1) * combination(8, 1) * combination(3, 1) = (7! / (1!(7-1)!)) * (8! / (1!(8-1)!)) * (3! / (1!(3-1)!)) = 7 * 8 * 3 = 168.
Step 3: Calculate the probability.
Probability = favorable outcomes / total outcomes = 168 / 816 ≈ 0.2059.
Therefore, the probability that one of each color is drawn is approximately 0.2059.