A florist has 2 azaleas, 3 buttercups, and 4 petunias. She puts two flowers together at random in a bouquet. However, the customer calls and says that she does not want two of the same flower. What is the probability that the florist does not have to change the bouquet?
Approach 1:
P(2 different flowers) = 1 - P(2 of the same flower).
P(2 azaleas):
P(1st flower is an azalea) = 2/9. (Of the 9 flowers, 2 are azaleas.)
P(2nd flower is an azalea) = 1/8. (Of the 8 remaining flowers, 1 is an azalea.)
Since we want both events to happen, we multiply the fractions:
2/9 * 1/8 = 1/36.
P(2 buttercups):
P(1st flower is a buttercup) = 3/9. (Of the 9 flowers, 3 are buttercups.)
P(2nd flower is a buttercup) = 2/8. (Of the 8 remaining flowers, 2 are buttercups.)
Since we want both events to happen, we multiply the fractions:
3/9 * 2/8 = 1/12.
P(2 petunias):
P(1st flower is a petunia) = 4/9. (Of the 9 flowers, 4 are petunias.)
P(2nd flower is a petunia) = 3/8. (Of the 8 remaining flowers, 3 are petunias.)
Since we want both events to happen, we multiply the fractions:
4/9 * 3/8 = 1/6.
Since any of the above outcomes would yield 2 of the same flower, we add the fractions:
P(2 of the same flower) = 1/36 + 1/12 + 1/6 = 10/36 = 5/18.
Thus, P(2 different flowers) = 1 - 5/18 = [spoiler]13/18[/spoiler].
Approach 2:
Case 1: Azalea on the first pick, different flower on the second pick
P(1st flower is an azalea) = 2/9. (Of the 9 flowers, 2 are azaleas.)
P(2nd flower is not an azalea) = 7/8. (Of the 8 remaining flowers, 7 are not azaleas.)
Since we want both events to happen, we multiply the fractions:
2/9 * 7/8 = 7/36.
Case 2: Buttercup on the first pick, different flower on the second pick
P(1st flower is a buttercup) = 3/9. (Of the 9 flowers, 3 are buttercups.)
P(2nd flower is not a buttercup) = 6/8. (Of the 8 remaining flowers, 6 are not buttercups.)
Since we want both events to happen, we multiply the fractions:
3/9 * 6/8 = 9/36.
Case 3: Petunia on the first pick, different flower on the second pick
P(1st flower is a petunia) = 4/9. (Of the 9 flowers, 4 are petunias.)
P(2nd flower is not a petunia) = 5/8. (Of the 8 remaining flowers, 5 are not petunias.)
Since we want both events to happen, we multiply the fractions:
4/9 * 5/8 = 10/36.
Since any of the above outcomes would yield 2 different types of flowers, we add the fractions:
7/36 + 9/36 + 10/36 = 26/36 = [spoiler]13/18[/spoiler].
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