A garden has a total of 10 flowers. Three are yellow, two are white and five are blue. If you select 2 at random without replacement, find the probability that:a. none of the flowers are yellow. b.at least one of the flowers is yellow.
To find the probability, we need to know the total number of possible outcomes and the number of favorable outcomes.
a. To find the probability that none of the flowers are yellow, we need to consider the total number of ways to select 2 flowers out of 10.
The total number of ways is the combination of 10 objects taken 2 at a time, which can be calculated using the formula:
C(n, r) = n! / (r!(n - r)!)
So, the total number of ways to select 2 flowers out of 10 is:
C(10, 2) = 10! / (2!(10-2)!) = 45
Now, let's look at the favorable outcomes, which means selecting two flowers that are not yellow. There are a total of 10 flowers, and 3 of them are yellow, so there are 7 flowers that are not yellow.
The total number of ways to select 2 flowers out of the 7 flowers that are not yellow is:
C(7, 2) = 7! / (2!(7-2)!) = 21
Therefore, the probability that none of the flowers are yellow is:
P(none are yellow) = favorable outcomes / total outcomes = 21 / 45 = 7 / 15
b. To find the probability that at least one of the flowers is yellow, we will find the complement of the event that none of the flowers are yellow.
Complement Rule: P(event A) = 1 - P(complement of A)
So, the probability that at least one flower is yellow is:
P(at least one yellow) = 1 - P(none are yellow)
We already calculated that P(none are yellow) is 7/15. Therefore,
P(at least one yellow) = 1 - (7/15) = 8/15