Problem Solving

VJesus12 wrote: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?

A. 5/18
B. 13/18
C. 1/9
D. 1/6
E. 2/9

First, we can REWRITE the question as "What is the probability that the two flowers are DIFFERENT colors?"
This is a good candidate to use the complement.
That is, P(Event A happening) = 1 - P(Event A not happening)

When applied to this questions, we get: P(DIFFERENT colors) = 1 - P(SAME color)

Aside: let A = azalea, let B = buttercup, let P = petunia

P(same color) = P(both A's OR both B's OR both P's)
= P(both A's) + P(both B's) + P(both P's)

Now let's examine each probability:

P(both A's)
We need the 1st flower to be an azalea and the 2nd flower to be an azalea
So, P(both A's) = (2/9)(1/8) = 2/72

P(both B's)
= (3/9)(2/8) = 6/72

P(both P's)
= (4/9)(3/8) = 12/72

So, P(same color) = (2/72) + (6/72) + (12/72) = 20/72 = 5/18

Now back to the beginning:
P(DIFFERENT colors) = 1 - P(SAME color)
= 1 - 5/18
= [spoiler]13/18 = B[/spoiler]

Cheers,
Brent

VJesus12 wrote: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?

A. 5/18
B. 13/18
C. 1/9
D. 1/6
E. 2/9

Here's an approach that does not use the complement.

To calculate P(2 diff colors), we need to consider 3 cases.
That is, P(2 diff colors) = P(azalea 1st then a different flower OR buttercup 1st then a different flower OR petunias 1st then a different flower)
= P(azalea 1st then a different flower) + P(buttercup 1st then a different flower) + P(petunia 1st then a different flower)

Let's examine each probability separately.

case 1: choose azalea first, choose different flower second
The probability of choosing an azalea first is 2/9
Once we have selected an azalea first, there are 8 flowers remaining (1 A, 3 B's and 4 P's), and we must choose a different flower second.
So, the probability of choosing a different flower second is 7/8
So, P(azalea first and different flower second) = (2/9)(7/8) = 14/72

case 2: choose buttercup first, choose different flower second
The probability of choosing an buttercup first is 3/9
Once we have selected an buttercup first, there are 8 flowers remaining (2 A's, 2 B's and 4 P's), and we must choose a different flower second.
So, the probability of choosing a different flower second is 6/8
So, P(buttercup first and different flower second) = (3/9)(6/8) = 18/72

case 3: choose petunia first, choose different flower second
The probability of choosing an petunia first is 4/9
Once we have selected an petunia first, there are 8 flowers remaining (2 A's, 3 B's and 3 P's), and we must choose a different flower second.
So, the probability of choosing a different flower second is 5/8
So, P(petunia first and different flower second) = (4/9)(5/8) = 20/72


P(2 diff colors) = (14/72) + (18/72) + (20/72)
= 52/72
= [spoiler]13/18 = B[/spoiler]

Cheers,
Brent

VJesus12 wrote: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?

A. 5/18
B. 13/18
C. 1/9
D. 1/6
E. 2/9

Calculating the probability that the florist does not have to change the bouquet is the same as calculating the probability that the bouquet has two different flowers. We can use the following formula:

1 = P(getting two of the same flower) + P(not getting two of the same flower)

Let's determine the probability of getting two of the same flower.

P(2 azaleas) = 2/9 x 1/8 = 2/72

P(2 buttercups) = 3/9 x 2/8 = 6/72

P(2 petunias) = 4/9 x 3/8 = 12/72

The probability of getting two of the same flower is, therefore, 2/72 + 6/72 + 12/72 = 20/72 = 5/18.

Thus, the probability of not getting two of the same flower is 1 - 5/18 = 13/18.

Answer: B