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A certain bag of gemstones is composed of two-thirds diamonds and one-third rubies. If the probability of randomly selecting two diamonds from the bag, without replacement, is 5/12, what is the probability of selecting two rubies from the bag, without replacement?
- a)5/36
- b)5/24
- c)1/12
- d)1/6
- e)1/4
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
A certain bag of gemstones is composed of two-thirds diamonds and one-...
The simplest way to solve the problem is to recognize that the total number of gems in the bag must be a multiple of 3, since we have 2/3 diamonds and 1/3 rubies. If we had a total number that was not divisible by 3, we would not be able to divide the stones into thirds. Given this fact, we can test some multiples of 3 to see whether any fit the description in the question. The smallest number of gems we could have is 6: 4 diamonds and 2 rubies (since we need at least 2 rubies). Is the probability of selecting two of these diamonds equal to 5/12? 4/6 × 3/5 = 12/30 = 2/5. Since this does not equal 5/12, this cannot be the total number of gems. The next multiple of 3 is 9, which yields 6 diamonds and 3 rubies: 6/9 × 5/8 = 30/72 = 5/12. Since this matches the probability in the question, we know we have 6 diamonds and 3 rubies. Now we can figure out the probability of selecting two rubies: 3/9 × 2/8 = 6/72 = 1/12
Most Upvoted Answer
A certain bag of gemstones is composed of two-thirds diamonds and one-...
To solve this problem, we can use the concept of probability and the fact that the bag is composed of two-thirds diamonds and one-third rubies.
Let's break down the problem step by step:
Step 1: Determine the probability of selecting two diamonds from the bag.
Since the bag is composed of two-thirds diamonds, the probability of selecting the first diamond is 2/3. After removing one diamond from the bag, there is only one diamond left out of the remaining gemstones. Therefore, the probability of selecting the second diamond, without replacement, is 1/2.
To find the probability of both events happening (selecting two diamonds), we multiply these probabilities together:
P(2 diamonds) = P(first diamond) * P(second diamond|first diamond)
= 2/3 * 1/2
= 1/3
Step 2: Determine the probability of selecting two rubies from the bag.
Since the bag is composed of one-third rubies, the probability of selecting the first ruby is 1/3. After removing one ruby from the bag, there is only one ruby left out of the remaining gemstones. Therefore, the probability of selecting the second ruby, without replacement, is 1/2.
To find the probability of both events happening (selecting two rubies), we multiply these probabilities together:
P(2 rubies) = P(first ruby) * P(second ruby|first ruby)
= 1/3 * 1/2
= 1/6
Step 3: Compare the probabilities of selecting two diamonds and two rubies.
We are given that the probability of selecting two diamonds is 5/12. Therefore, we can set up the following equation:
P(2 diamonds) = 5/12
Substituting the value of P(2 diamonds) from Step 1, we have:
1/3 = 5/12
To find the probability of selecting two rubies, we can set up the following equation:
P(2 rubies) = x
Substituting the value of P(2 rubies) from Step 2, we have:
1/6 = x
Therefore, the probability of selecting two rubies from the bag, without replacement, is 1/6, which corresponds to option C.
Let's break down the problem step by step:
Step 1: Determine the probability of selecting two diamonds from the bag.
Since the bag is composed of two-thirds diamonds, the probability of selecting the first diamond is 2/3. After removing one diamond from the bag, there is only one diamond left out of the remaining gemstones. Therefore, the probability of selecting the second diamond, without replacement, is 1/2.
To find the probability of both events happening (selecting two diamonds), we multiply these probabilities together:
P(2 diamonds) = P(first diamond) * P(second diamond|first diamond)
= 2/3 * 1/2
= 1/3
Step 2: Determine the probability of selecting two rubies from the bag.
Since the bag is composed of one-third rubies, the probability of selecting the first ruby is 1/3. After removing one ruby from the bag, there is only one ruby left out of the remaining gemstones. Therefore, the probability of selecting the second ruby, without replacement, is 1/2.
To find the probability of both events happening (selecting two rubies), we multiply these probabilities together:
P(2 rubies) = P(first ruby) * P(second ruby|first ruby)
= 1/3 * 1/2
= 1/6
Step 3: Compare the probabilities of selecting two diamonds and two rubies.
We are given that the probability of selecting two diamonds is 5/12. Therefore, we can set up the following equation:
P(2 diamonds) = 5/12
Substituting the value of P(2 diamonds) from Step 1, we have:
1/3 = 5/12
To find the probability of selecting two rubies, we can set up the following equation:
P(2 rubies) = x
Substituting the value of P(2 rubies) from Step 2, we have:
1/6 = x
Therefore, the probability of selecting two rubies from the bag, without replacement, is 1/6, which corresponds to option C.
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