A bag contains three green Christmas ornaments and four gold ornaments. If you randomly pick two

ornaments from the bag, at the same time, what is the probability that both ornaments will be gold?

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To find the probability that both ornaments will be gold, we need to calculate the probability of selecting a gold ornament on the first pick, and then the probability of selecting a gold ornament on the second pick.

Step 1: Calculate the probability of selecting a gold ornament on the first pick.
There are a total of 3 + 4 = 7 ornaments in the bag, and 4 of them are gold.
Therefore, the probability of selecting a gold ornament on the first pick is 4/7.

Step 2: Calculate the probability of selecting a gold ornament on the second pick after selecting a gold ornament on the first pick.
After the first pick, there will be a total of 6 ornaments left in the bag, with 3 gold ornaments remaining.
Therefore, the probability of selecting a gold ornament on the second pick after selecting a gold ornament on the first pick is 3/6.

Step 3: Multiply the probabilities from Step 1 and Step 2.
To find the probability of both events occurring (selecting a gold ornament on the first pick and then a gold ornament on the second pick), we multiply the probabilities:
Probability = (4/7) * (3/6) = 12/42 = 2/7.

Therefore, the probability that both ornaments will be gold is 2/7.

To find the probability of picking two gold ornaments, we need to determine the total number of possible outcomes and the number of favorable outcomes.

First, let's calculate the total number of possible outcomes:
When picking two ornaments, there are a total of C(7, 2) possible outcomes. The combination formula is given by C(n, r) = n! / (r! * (n-r)!), where n is the total number of items and r is the number of items to be selected. Therefore, C(7, 2) = 7! / (2! * (7-2)!) = 21.

Next, let's determine the number of favorable outcomes:
Since we need to pick two gold ornaments, there are a total of C(4, 2) favorable outcomes. C(4, 2) = 4! / (2! * (4-2)!) = 6.

Finally, we can calculate the probability:
The probability of picking two gold ornaments is the number of favorable outcomes divided by the total number of possible outcomes.
So, the probability is P = C(4, 2) / C(7, 2) = 6 / 21 = 2 / 7.

Therefore, the probability that both ornaments will be gold is 2/7 or approximately 0.2857 (rounded to four decimal places).

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