A bottle contains 12 red marbles and 8 blue marbles. A marble is chosen at random and not replaced. Then, a second marble is chosen at random. Determine the probability that the two marbles are not the same color. Determine the probability that at least one of the marbles is red.
20 marbles in all
P1(red,blue) = 12/20 * 8/19
P2(blue,red) = 8/20 * 12/19
so, P(different) = P1+P2 = 48/95
Note that
P1(red,red) = 12/20 * 11/19
P2(blue,blue) = 8/20 * 7/19
P(not same) = 1 - (P1+P2) = 48/95
P(at least 1 red) = 1 - P(blue,blue) = 81/95
Check to see that this is also P(not same) + P(red,red)
Well, well, well. Looks like we've got a colorful situation here with marbles! Let's get into it, shall we?
To find the probability that the two marbles are not the same color, we can consider the two cases separately:
Case 1: The first marble is red and the second marble is blue.
The probability of selecting a red marble first is 12/20, and after that, the probability of selecting a blue marble is 8/19. So, the probability for Case 1 is (12/20) * (8/19).
Case 2: The first marble is blue and the second marble is red.
The probability of selecting a blue marble first is 8/20, and after that, the probability of selecting a red marble is 12/19. So, the probability for Case 2 is (8/20) * (12/19).
To find the overall probability, we sum up the probabilities of the two cases:
(12/20) * (8/19) + (8/20) * (12/19) = 96/380 + 96/380 = 192/380
Simplifying the fraction, we get 24/47 or approximately 0.51.
So, the probability that the two marbles are not the same color is approximately 0.51 or 51%.
Now onto the second question.
To find the probability that at least one of the marbles is red, we can think of the complementary event where neither of the marbles is red. This means both marbles have to be blue.
The probability of selecting a blue marble first is 8/20, and after that, the probability of selecting another blue marble is 7/19. So, the probability of getting two blue marbles is (8/20) * (7/19).
To find the probability of at least one red marble, we subtract the probability of two blue marbles from 1 (since it's the complement):
1 - (8/20) * (7/19) = 1 - 56/380 = 1 - 14/95
Simplifying the fraction, we get 81/95 or approximately 0.85.
So, the probability that at least one of the marbles is red is approximately 0.85 or 85%.
That's the colorful world of marble probabilities for you!
To determine the probability that the two marbles are not the same color, we need to consider two scenarios: one where the first marble chosen is red and the second marble is blue, and another where the first marble chosen is blue and the second marble is red.
First, let's calculate the probability of selecting a red marble first:
Probability of selecting a red marble = Number of red marbles / Total number of marbles
= 12 / (12 + 8)
= 12 / 20
= 3 / 5
After selecting a red marble first, there are now 11 red marbles and 8 blue marbles left in the bottle. Consequently, for the second marble to be blue, the probability can be calculated as:
Probability of selecting a blue marble second (after the first marble is red) = Number of blue marbles / Total number of marbles left
= 8 / (11 + 8)
= 8 / 19
Now, let's calculate the probability of selecting a blue marble first:
Probability of selecting a blue marble = Number of blue marbles / Total number of marbles
= 8 / (12 + 8)
= 8 / 20
= 2 / 5
After selecting a blue marble first, there are now 12 red marbles and 7 blue marbles left in the bottle. Consequently, for the second marble to be red, the probability can be calculated as:
Probability of selecting a red marble second (after the first marble is blue) = Number of red marbles / Total number of marbles left
= 12 / (12 + 7)
= 12 / 19
The total probability of the two marbles not being the same color is the sum of the probability of getting a red marble first and a blue marble second, and the probability of getting a blue marble first and a red marble second:
Probability of not the same color = (Probability of red first and blue second) + (Probability of blue first and red second)
= (3/5) * (8/19) + (2/5) * (12/19)
= (24/95) + (24/95)
= 48/95
≈ 0.5053
Therefore, the probability that the two marbles are not the same color is approximately 0.5053.
To determine the probability that at least one of the marbles is red, we can consider two scenarios: one where the two marbles are the same color (both red or both blue) and another where the two marbles are not the same color.
We have already determined the probability of the two marbles not being the same color as approximately 0.5053. Therefore, the probability of the two marbles being the same color is 1 minus the probability of them not being the same color:
Probability of two marbles being the same color = 1 - Probability of not the same color
= 1 - 48/95
= 47/95
≈ 0.4947
Finally, the probability that at least one of the marbles is red can be calculated as the complement event of both marbles being the same color:
Probability of at least one red marble = 1 - Probability of two marbles being the same color
= 1 - 47/95
= 48/95
≈ 0.5053
Therefore, the probability that at least one of the marbles is red is approximately 0.5053.
To summarize:
- The probability that the two marbles are not the same color is approximately 0.5053.
- The probability that at least one of the marbles is red is also approximately 0.5053.
To determine the probability that the two marbles are not the same color, we need to consider two scenarios: one where the first marble chosen is red and the second is blue, and another where the first marble chosen is blue and the second is red.
First, let's calculate the probability of drawing a red marble and then a blue marble.
The probability of drawing a red marble first is given by the ratio of the number of red marbles to the total number of marbles: 12 red marbles / (12 red marbles + 8 blue marbles) = 12/20 = 3/5.
After drawing the first red marble, there are now 11 red marbles and 8 blue marbles remaining in the bottle. The probability of drawing a blue marble second is given by the ratio of the number of blue marbles to the total remaining number of marbles: 8 blue marbles / (11 red marbles + 8 blue marbles) = 8/19.
The probability of drawing a red marble first and then a blue marble is the product of these probabilities: (3/5) * (8/19) = 24/95.
Similarly, the probability of drawing a blue marble first and then a red marble can be calculated as follows:
The probability of drawing a blue marble first is given by the ratio of the number of blue marbles to the total number of marbles: 8 blue marbles / (12 red marbles + 8 blue marbles) = 8/20 = 2/5.
After drawing the first blue marble, there are now 12 red marbles and 7 blue marbles remaining in the bottle. The probability of drawing a red marble second is given by the ratio of the number of red marbles to the total remaining number of marbles: 12 red marbles / (12 red marbles + 7 blue marbles) = 12/19.
The probability of drawing a blue marble first and then a red marble is the product of these probabilities: (2/5) * (12/19) = 24/95.
Therefore, the probability that the two marbles are not the same color is the sum of these two probabilities: 24/95 + 24/95 = 48/95.
Now, let's calculate the probability that at least one of the marbles is red.
To find this probability, we can use the concept of the complement. The complement of the event "at least one marble is red" is the event "no marbles are red."
The probability of drawing no red marbles can be calculated by finding the ratio of the number of blue marbles to the total number of marbles: 8 blue marbles / (12 red marbles + 8 blue marbles) = 8/20 = 2/5.
Therefore, the probability that at least one marble is red is 1 - (2/5) = 3/5.