There are 6 people in a gardening club. Each gardener orders seeds from a list of 11 different types of available seeds. What is the probability that at least 2 gardeners order the same type of seeds?
To calculate the probability that at least 2 gardeners order the same type of seeds, we can take the opposite approach and calculate the probability that all gardeners order different types of seeds.
First, let's consider the first gardener. They have a choice of all 11 different types of seeds. Therefore, the probability that the first gardener chooses a unique type of seed is 1.
Now, for the second gardener, there are only 10 remaining types of seeds to choose from (since the first gardener has already chosen one). So, the probability that the second gardener chooses a unique type of seed is 10/11.
For the third gardener, there are 9 remaining types of seeds to choose from, and their probability of choosing a unique type is 9/11.
We continue this pattern for each gardener, decrementing the number of available types of seeds by 1 each time. So, the probability that all 6 gardeners choose unique types of seeds is:
1 * (10/11) * (9/11) * (8/11) * (7/11) * (6/11)
Calculating this expression:
(10/11) * (9/11) * (8/11) * (7/11) * (6/11) ≈ 0.3395
Therefore, the probability that at least 2 gardeners order the same type of seeds is the complement of the above probability:
1 - 0.3395 ≈ 0.6605
So, the probability that at least 2 gardeners order the same type of seeds is approximately 0.6605, or 66.05%.
To find the probability that at least 2 gardeners order the same type of seeds, we can use the concept of complementary probability.
First, let's find the probability that no two gardeners order the same type of seeds.
For the first gardener, there are 11 options to choose from.
For the second gardener, there are 10 remaining options, as one type has already been chosen by the first gardener.
For the third gardener, there are 9 remaining options, and so on.
So, the probability that no two gardeners order the same type of seeds can be calculated as follows:
P(no matches) = (11/11) * (10/11) * (9/11) * (8/11) * (7/11) * (6/11)
Now, we can find the complementary probability, which is the probability that at least 2 gardeners order the same type of seeds.
P(at least 2 matches) = 1 - P(no matches)
P(at least 2 matches) = 1 - [(11/11) * (10/11) * (9/11) * (8/11) * (7/11) * (6/11)]
Simplifying the expression will give us the probability.
P(at least 2 matches) = 1 - (0.5214...)
P(at least 2 matches) ≈ 0.4785
Therefore, the probability that at least 2 gardeners order the same type of seeds is approximately 0.4785 or 47.85%.