Question
A big flower vase has 5 begonias, 6 carnations, and 7 tulips. Two flowers are randomly selected (without replacement). What is the probability of selecting two tulips? (Express all answer in the simplest form of a fraction.) What is the probability of selecting the first tulip? What is the probability of selecting the second tulip?
Answers
Answered by
GPT 3.5
To calculate the probability of selecting two tulips, first we need to find the total number of ways to choose 2 flowers from the 18 flowers in the vase.
The total number of ways to choose 2 flowers from 18 flowers is given by the combination formula:
$${{n}\choose{k}} = \frac{n!}{k!(n-k)!}$$
Therefore, the total number of ways to choose 2 flowers from 18 flowers is:
$${{18}\choose{2}} = \frac{18!}{2!(18-2)!} = 153$$
Now we need to find the number of ways to choose 2 tulips out of the 7 tulips in the vase. This can be done using the combination formula:
$${{7}\choose{2}} = \frac{7!}{2!(7-2)!} = 21$$
So, the probability of selecting two tulips is:
$$P(\text{two tulips}) = \frac{21}{153} = \frac{7}{51}$$
Next, to find the probability of selecting the first tulip, we consider that after selecting the first tulip, there are 17 flowers left in the vase, and 6 of them are tulips. Therefore, the probability of selecting the first tulip is:
$$P(\text{first tulip}) = \frac{7}{18}$$
Finally, to find the probability of selecting the second tulip, after selecting the first tulip, there are now 17 flowers in the vase and only 6 of them are tulips. Therefore, the probability of selecting the second tulip given that the first was a tulip is:
$$P(\text{second tulip}) = \frac{6}{17}$$
The total number of ways to choose 2 flowers from 18 flowers is given by the combination formula:
$${{n}\choose{k}} = \frac{n!}{k!(n-k)!}$$
Therefore, the total number of ways to choose 2 flowers from 18 flowers is:
$${{18}\choose{2}} = \frac{18!}{2!(18-2)!} = 153$$
Now we need to find the number of ways to choose 2 tulips out of the 7 tulips in the vase. This can be done using the combination formula:
$${{7}\choose{2}} = \frac{7!}{2!(7-2)!} = 21$$
So, the probability of selecting two tulips is:
$$P(\text{two tulips}) = \frac{21}{153} = \frac{7}{51}$$
Next, to find the probability of selecting the first tulip, we consider that after selecting the first tulip, there are 17 flowers left in the vase, and 6 of them are tulips. Therefore, the probability of selecting the first tulip is:
$$P(\text{first tulip}) = \frac{7}{18}$$
Finally, to find the probability of selecting the second tulip, after selecting the first tulip, there are now 17 flowers in the vase and only 6 of them are tulips. Therefore, the probability of selecting the second tulip given that the first was a tulip is:
$$P(\text{second tulip}) = \frac{6}{17}$$