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?
the first gardener can pick from 11 different seeds, the second can pick from 11 different seeds, etc
(this allows the same seed being picked by more than one gardener)
= 11^6 ways = 1771561
number of ways where none are the same, (once picked, can't be picked again)
= 11x10x9x8x7x6 =332640
so the number where at least 2 are the same
= 1771561 - 332640 = 1438921
so prob(at least 2 the same) = 1438921/1771561 = .8122
check my arithmetic, it is often quite lousy
A fair coin is tossed 4 times. Given that each of the first 3 tosses land tails up, what is the probability that all 4 tosses land tails up?
Well, probability is a funny thing, just like a clown at a garden party! Let's calculate it step by step.
First, let's say Gardener 1 orders their seeds. They have 11 options to choose from. Now, Gardener 2 comes along and orders their seeds. There's a 1 in 11 chance they choose the same type as Gardener 1.
Moving on to Gardener 3, the probability gets a little trickier. They have 11 options as well, but now they need to avoid both Gardener 1 and Gardener 2's choices. So, the chances they order a different seed are 10 out of 11.
For Gardener 4, they must avoid the choices of Gardener 1, 2, and 3. So the probability of them ordering a different seed is 9 out of 11.
Following the same pattern, the probability for Gardener 5 is 8 out of 11, and for Gardener 6, it's 7 out of 11.
Now let's bring all these probabilities together. In order for at least 2 gardeners to choose the same type of seeds, we need the probability of them not choosing the same type to be 0. So, we multiply all the probabilities together:
(1/11) * (10/11) * (9/11) * (8/11) * (7/11).
Calculating this, we find that the probability is approximately 0.0452, or about 4.52%.
So, there's a 4.52% chance that at least 2 gardeners in the club will order the same type of seeds. It's not too likely, but hey, in the world of probability, anything can happen!
To calculate the probability that at least 2 gardeners order the same type of seeds, we can make use of the concept of complement probability.
First, let's calculate the probability that no two gardeners order the same type of seeds. For the first gardener, there are 11 types of seeds to choose from. For the second gardener, there are 10 remaining types of seeds to choose from since we want to ensure there are no duplicates. Similarly, the third gardener has 9 types of seeds to choose from, the fourth gardener has 8 types, the fifth gardener has 7 types, and the sixth gardener has 6 types.
Now, multiplying these probabilities together, we get:
(11/11) × (10/11) × (9/11) × (8/11) × (7/11) × (6/11) = 0.286
However, we are interested in the complement, which is the probability of two or more gardeners ordering the same type of seeds. So, we subtract this value from 1:
1 - 0.286 = 0.714
Therefore, the probability that at least 2 gardeners order the same type of seeds is approximately 0.714, or 71.4%.
Note: This calculation assumes that each gardener chooses the type of seed independently and with equal probability.