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?

Answer 1

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:

$$P(\text{second tulip}) = \frac{6}{17}$$