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?
To find the probability, we need to determine the number of favorable outcomes (both ornaments are gold) and divide it by the number of total possible outcomes.
There are 4 gold ornaments in the bag. When we randomly select the first one, there are 4 gold ornaments out of a total of 7 ornaments.
After selecting the first gold ornament, there are now 3 gold ornaments remaining out of a total of 6 remaining ornaments. Thus, the probability of selecting a second gold ornament after the first one is 3/6.
To find the overall probability, we multiply the probabilities of each event happening together. Therefore, the probability of picking two gold ornaments from the bag is (4/7) * (3/6) = 12/42 = 2/7 ≈ 0.2857.
So, the probability that both ornaments will be gold is approximately 0.2857 or 28.57%.
To find the probability of picking two gold ornaments, we need to calculate the probability of the first ornament being gold and the second ornament being gold, assuming that the first ornament is not replaced.
Step 1: Calculate the probability of picking the first gold ornament:
There are a total of seven ornaments in the bag, and four of them are gold. Therefore, the probability of picking a gold ornament as the first one is 4/7.
Step 2: Calculate the probability of picking the second gold ornament:
After removing one gold ornament from the bag, there are now three gold ornaments left and a total of six ornaments. Therefore, the probability of picking a gold ornament as the second one, given that the first one was gold and not replaced, is 3/6.
Step 3: Multiply the probabilities:
To calculate the probability of both ornaments being gold, we multiply the probability of the first ornament being gold (4/7) by the probability of the second ornament being gold (3/6).
(4/7) * (3/6) = 12/42 = 6/21 = 2/7.
Therefore, the probability that both ornaments will be gold is 2/7.