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Bunuel
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Hi Bunel,

I thought :
Probability of not picking two tulips :
6/8*5/7=5/12

I know I didn't get the correct answer but why do I have to find first the probability of getting both tulips (as you suggest)?
Could you please explain me ?

Thanks,
Daniela

Notice that we are allowed to have one tulip, while you are calculating the probability of 0 tulips.

P(not 2 tulips) = P(1 tulip) + P(0 tulips) = 2*2/8*6/7 + 6/8*5/7 = 27/28.
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2 tulips out of 8 flowers can be selected by 1 way.

Total No. of ways 2 flowers can be selected out of 8 flowers is 8c2= 7*8/2= 28

Probability of selecting 2 tulips out of 8 flowers is 1/28.

Therefor P(Not Selecting a Tulip)= 1-(1/28)= 28-1/28=27/28
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Hi Bunnel,

Can we take here for non relevant event (Not selecting the tulip) 6C6

and then find the P = 1- (Non relevant event/relevant event) = 1-(6C6/8C2) = 1-(1/28). Is it correct approach ?

Thanks,
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swapnils10
Hi Bunnel,

Can we take here for non relevant event (Not selecting the tulip) 6C6

and then find the P = 1- (Non relevant event/relevant event) = 1-(6C6/8C2) = 1-(1/28). Is it correct approach ?

Thanks,

It should be:

\(P(no \ two \ tulips) = 1 - P(two \ tulips) = 1 - \frac{the \ number \ of \ ways \ to \ pick \ two \ tulips \ out \ of \ two}{the \ number \ of \ ways \ to \ pick \ two \ flowers \ out \ of \ eight}= 1 - \frac{C^2_2}{C^2_8}=\frac{27}{28}\).
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
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Bunuel,

I have a doubt w.r.t the total possible outcomes.

While, mathematically it makes sense to take 8C2 = 28 ways as the total, but how would it look like if one were to list down all possible outcomes?

We have 2 Roses, 2 Lilies, 2 Daisies, and 2 Tulips. Total flowers = 2R, 2L, 2D, 2T.

While listing down, I could only list 10 combinations.
RL, RD, RT, RR
LD, LT, LL
DT, DD
TT

Since we are selecting, so order shouldn't matter, and since the 2 Rs / Ls / Ds / Ts are similar to each other, so shouldn't the option R1 L be the same as the option R2 L?

When I counted the 2 similar flowers as separate options, then the outcomes are:

R1 L1 R1 D1 R1 T1R1 L2R1 D2 R1 T2 R1 R2
R2 L1 R2 D1 R2 T1 R2 L2 R2 D2R2 T2
L1 T1 L1 D1 L1 T2 L2 D2L1 L2
L2 T1 L2 D1 L2 T2 L2 D2
D1 T1 D1 T2 D1 D2
D2 T1 D2 T2
T1 T2

Here the total comes to 28 and matches the result from the formula. But then the flowers, even though similar, are supposed to be counted separately. Why so?
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Bunuel
Official Solution:

A flower shop has 2 tulips, 2 roses, 2 daisies, and 2 lilies. If two flowers are chosen at random one after the other without replacement, what is the probability of not selecting exactly two tulips?

A. \(\frac{1}{8}\)
B. \(\frac{1}{7}\)
C. \(\frac{1}{2}\)
D. \(\frac{7}{8}\)
E. \(\frac{27}{28}\)


Let's find the probability of the opposite event, which is choosing two tulips, and subtract that from 1. The probability of choosing two tulips is \(P(both \ tulips) = \frac{2}{8}*\frac{1}{7} = \frac{1}{28}\).

Therefore, the probability that not both selected flowers are tulips is \(P(not \ both \ are \ tulips) = 1 - P(both \ tulips) = 1 - \frac{1}{28} = \frac{27}{28}\).


Answer: E

Bunuel,

I have a doubt w.r.t the total possible outcomes.

While, mathematically it makes sense to take 8C2 = 28 ways as the total, but how would it look like if one were to list down all possible outcomes?

We have 2 Roses, 2 Lilies, 2 Daisies, and 2 Tulips. Total flowers = 2R, 2L, 2D, 2T.

While listing down, I could only list 10 combinations.
RL, RD, RT, RR
LD, LT, LL
DT, DD
TT

Since we are selecting, so order shouldn't matter, and since the 2 Rs / Ls / Ds / Ts are similar to each other, so shouldn't the option R1 L be the same as the option R2 L?

When I counted the 2 similar flowers as separate options, then the outcomes are:

R1 L1 R1 D1 R1 T1R1 L2R1 D2 R1 T2 R1 R2
R2 L1 R2 D1 R2 T1 R2 L2 R2 D2R2 T2
L1 T1 L1 D1 L1 T2 L2 D2L1 L2
L2 T1 L2 D1 L2 T2 L2 D2
D1 T1 D1 T2 D1 D2
D2 T1 D2 T2
T1 T2

Here the total comes to 28 and matches the result from the formula. But then the flowers, even though similar, are supposed to be counted separately. Why so?

Why use 8C2 = 28 as total outcomes?

Because we are selecting 2 flowers out of 8 distinct positions.

Even though the flowers look alike (e.g., both tulips), they are still are physically distinct objects.

For example:

Tulip1 ≠ Tulip2, even if both are tulips.

So the set of 8 flowers is: T1, T2, R1, R2, D1, D2, L1, L2, so 8 distinct entities. Choosing any 2 of these gives 8C2 = 28 combinations.
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I like the solution - it’s helpful.
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