Welcome to Westonci.ca, where finding answers to your questions is made simple by our community of experts. Discover a wealth of knowledge from professionals across various disciplines on our user-friendly Q&A platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
To find the probability that one marble is yellow and the other marble is red when two marbles are chosen from a bag that contains eight yellow marbles, nine green marbles, three purple marbles, and five red marbles, we need to proceed as follows:
1. Calculate the total number of marbles in the bag:
[tex]\[ \text{Total marbles} = 8 (\text{yellow}) + 9 (\text{green}) + 3 (\text{purple}) + 5 (\text{red}) = 25 \][/tex]
2. Calculate the probability of drawing one yellow marble and then one red marble:
- First, the probability of drawing a yellow marble:
[tex]\[ P(\text{Yellow first}) = \frac{8}{25} \][/tex]
- After one yellow marble is drawn, there are now 24 marbles left, including 5 red marbles. The probability of drawing a red marble next:
[tex]\[ P(\text{Red second | Yellow first}) = \frac{5}{24} \][/tex]
- Therefore, the combined probability for this scenario is:
[tex]\[ P(\text{Yellow first and Red second}) = \frac{8}{25} \times \frac{5}{24} \][/tex]
3. Calculate the probability of drawing one red marble and then one yellow marble:
- First, the probability of drawing a red marble:
[tex]\[ P(\text{Red first}) = \frac{5}{25} \][/tex]
- After one red marble is drawn, there are now 24 marbles left, including 8 yellow marbles. The probability of drawing a yellow marble next:
[tex]\[ P(\text{Yellow second | Red first}) = \frac{8}{24} \][/tex]
- Therefore, the combined probability for this scenario is:
[tex]\[ P(\text{Red first and Yellow second}) = \frac{5}{25} \times \frac{8}{24} \][/tex]
4. Combine both scenarios:
Since the two events are mutually exclusive (they don't overlap), the overall probability is the sum of both combined probabilities:
[tex]\[ P(\text{One Yellow and One Red}) = P(\text{Yellow first and Red second}) + P(\text{Red first and Yellow second}) \][/tex]
[tex]\[ P(\text{One Yellow and One Red}) = \left( \frac{8}{25} \times \frac{5}{24} \right) + \left( \frac{5}{25} \times \frac{8}{24} \right) \][/tex]
5. Simplify the expression:
Each term in the sum simplifies to:
[tex]\[ \frac{8 \times 5}{25 \times 24} + \frac{5 \times 8}{25 \times 24} = 2 \times \frac{8 \times 5}{25 \times 24} = 2 \times \frac{40}{600} = 2 \times \frac{1}{15} = \frac{2}{15} \][/tex]
Therefore, the expression that gives the probability that one marble is yellow and the other marble is red is:
[tex]\[ P(\text{Yellow and Red}) = \frac{2}{15} \][/tex]
Given the options provided in the original question, the closest representation of this result is:
[tex]\[ P(\text{Y and R}) = \frac{\left(C_1 C_8 \right) \left(C_1 \right)}{{ }_2 C_{25}} \][/tex]
This matches with the simplification process described and indicates the correct answer is:
[tex]\[ \mathbb{P}(Y \text{ and } R) = \frac{8}{25} \times \frac{5}{24} + \frac{5}{25} \times \frac{8}{24} = \left(\frac{8}{25} \cdot \frac{5}{24}\right) + \left(\frac{5}{25} \cdot \frac{8}{24}\right) = 2 \left(\frac{8}{25} \cdot \frac{5}{24}\right) = \frac{2}{15} \][/tex]
1. Calculate the total number of marbles in the bag:
[tex]\[ \text{Total marbles} = 8 (\text{yellow}) + 9 (\text{green}) + 3 (\text{purple}) + 5 (\text{red}) = 25 \][/tex]
2. Calculate the probability of drawing one yellow marble and then one red marble:
- First, the probability of drawing a yellow marble:
[tex]\[ P(\text{Yellow first}) = \frac{8}{25} \][/tex]
- After one yellow marble is drawn, there are now 24 marbles left, including 5 red marbles. The probability of drawing a red marble next:
[tex]\[ P(\text{Red second | Yellow first}) = \frac{5}{24} \][/tex]
- Therefore, the combined probability for this scenario is:
[tex]\[ P(\text{Yellow first and Red second}) = \frac{8}{25} \times \frac{5}{24} \][/tex]
3. Calculate the probability of drawing one red marble and then one yellow marble:
- First, the probability of drawing a red marble:
[tex]\[ P(\text{Red first}) = \frac{5}{25} \][/tex]
- After one red marble is drawn, there are now 24 marbles left, including 8 yellow marbles. The probability of drawing a yellow marble next:
[tex]\[ P(\text{Yellow second | Red first}) = \frac{8}{24} \][/tex]
- Therefore, the combined probability for this scenario is:
[tex]\[ P(\text{Red first and Yellow second}) = \frac{5}{25} \times \frac{8}{24} \][/tex]
4. Combine both scenarios:
Since the two events are mutually exclusive (they don't overlap), the overall probability is the sum of both combined probabilities:
[tex]\[ P(\text{One Yellow and One Red}) = P(\text{Yellow first and Red second}) + P(\text{Red first and Yellow second}) \][/tex]
[tex]\[ P(\text{One Yellow and One Red}) = \left( \frac{8}{25} \times \frac{5}{24} \right) + \left( \frac{5}{25} \times \frac{8}{24} \right) \][/tex]
5. Simplify the expression:
Each term in the sum simplifies to:
[tex]\[ \frac{8 \times 5}{25 \times 24} + \frac{5 \times 8}{25 \times 24} = 2 \times \frac{8 \times 5}{25 \times 24} = 2 \times \frac{40}{600} = 2 \times \frac{1}{15} = \frac{2}{15} \][/tex]
Therefore, the expression that gives the probability that one marble is yellow and the other marble is red is:
[tex]\[ P(\text{Yellow and Red}) = \frac{2}{15} \][/tex]
Given the options provided in the original question, the closest representation of this result is:
[tex]\[ P(\text{Y and R}) = \frac{\left(C_1 C_8 \right) \left(C_1 \right)}{{ }_2 C_{25}} \][/tex]
This matches with the simplification process described and indicates the correct answer is:
[tex]\[ \mathbb{P}(Y \text{ and } R) = \frac{8}{25} \times \frac{5}{24} + \frac{5}{25} \times \frac{8}{24} = \left(\frac{8}{25} \cdot \frac{5}{24}\right) + \left(\frac{5}{25} \cdot \frac{8}{24}\right) = 2 \left(\frac{8}{25} \cdot \frac{5}{24}\right) = \frac{2}{15} \][/tex]
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.