I07-17

Question

Every New Year's Eve, Pratik gives scarves as gifts to his coworkers. This year, to show his appreciation, he asks Malachi to be the first to choose three scarves. To make it more fun, Malachi must pick three scarves blindly from a box containing 10 scarves.
 
The box holds 3 grey, 4 white, 2 yellow, and 1 black scarf. In the table, select one value for the probability that Malachi picks  at least one grey scarf  and another for the probability of picking  exactly two grey scarves . Make only two selections, one in each column.

Bunuel · Answer

Official Solution:

To find the probability of Malachi picking  at least one grey scarf , let's calculate the probability of the opposite event and subtract it from 1. The opposite event is selecting zero grey scarves, meaning choosing all three scarves from the 4 white, 2 yellow, and 1 black scarf. Therefore, the probability is  P(at \ least \ one \ grey \ scarf) = 1 - \frac{C^3_7}{C^3_{10}}= 1 - \frac{35}{120} = \frac{17}{24} .
 
Alternatively, we could have used a probability approach in the last step:  P(at \ least \ one \ grey \ scarf) = 1 - \frac{7}{10} * \frac{6}{9} * \frac{5}{8} = \frac{17}{24} .
 
To find the probability of Malachi picking  exactly two grey scarves , we calculate the probability of selecting two grey scarves and one scarf from the 7 non-grey scarves:  P(exactly \ two \ grey \ scarves) = \frac{C^2_3 * C^1_7}{C^3_{10}}= \frac{3 * 7}{120} = \frac{7}{40} .

Correct answer:

At least one grey scarf  " \frac{17}{24} "

Exactly two grey scarves  " \frac{7}{40} "

Bunuel · Answer

I have revised the question, solution, and formatting by adding more details to enhance clarity

Soga15 · Answer

Hi! I understand the first solution. However for the second one I am getting 7/120 and would like to know if someone can help me here. I have the following as solution:
3/10 * 2/9 * 7/8 = 7/120

Thanks!

Bunuel · Answer

That's a common mistake when solving with the probability approach. The point is that picking exactly two grey scarves, or the GGX scenario, can occur in three different ways: GGX, GXG, and XGG. Each has a probability of 7/120, making the overall probability equal to 7/120 * 3 = 7/40.

_sahuji_ · Answer

I did not quite understand the solution. Why we are considering scarves as different objects. Shouldn't it be a case of identical objects. So, it doesn't matter how the combination should be.

Bunuel · Answer

Scarves of the same color are still physically different items, so we treat them as distinct when calculating combinations.

Vedant41 · Answer

Hi  Bunuel ,

The mistake that I made in the second statement was that, I used the combinations formula to calculate 3C2 and & 7C1. Then I multiplied the entire fraction 21/120 by 3. Because the selected entities from both Grey and other colors, can be arranged in 3 ways.

Can you please clarify why this approach is incorrect and during which situations is it correct to consider permutations and multiply with the extra number of arrangements?

Bunuel · Answer

Are you talking about “at least one grey scarf”? Can you please clarify how exactly what you wrote accounts for that?

Vedant41 · Answer

I was talking about the second solution that we need to find " picking exactly two grey scarves". For calculating solution for that statement, I used the combinations formula to calculate 3C2 (selecting 2 grey scarves from 3) and & 7C1 (selecting 1 from the remaining 7). Then I multiplied the entire fraction (21/120) by 3. Because I thought that the selected entities from both Grey and other colors, can be arranged in 3 ways (For eg- GGO (O- other scarves), GOG, OGG)

Can you please clarify why this approach is incorrect and during which situations is it correct to consider permutations and multiply with the extra number of arrangements?

Can you please clarify why this approach is incorrect and during which situations is it correct to consider permutations and multiply with the extra number of arrangements?

Bunuel · Answer

The key here is maintaining consistency between the numerator and denominator, and understanding what multiplying by 3!/2! actually does.

We multiply 3/10 * 2/9 * 7/8 by 3!/2! because the GGO outcome can occur in 3 different ways:

GGO (first scarf is grey, second is grey, third is other) 
 GOG (first scarf is grey, second is other, third is grey) 
 OGG (first scarf is other, second is grey, third is grey)

These are three different scenarios, and we need to account for all of them when calculating the probability.

Alternatively, we could calculate:

P(GGO) + P(GOG) + P(OGG) =
= (3/10 * 2/9 * 7/8) + (3/10 * 7/9 * 2/8) + (7/10 * 3/9 * 2/8)
= (3/10 * 2/9 * 7/8) * 3

Notice that both approaches yield the same result: (3/10 * 2/9 * 7/8) * 3

Now, when using the combinations approach, we calculate 3C2 * 7C1 / 10C3, where the numerator represents the number of different groups of three scarves where 2 are grey and 1 is other, and the denominator represents the total number of ways to select 3 scarves from 10. It's crucial to observe that both the numerator and the denominator give unordered groups of three scarves (meaning XYZ is counted only once, without considering XZY, YXZ, and other arrangements). This maintains consistency between them.

Therefore, multiplying the numerator by 3!/2!, which accounts for arrangements, would break the consistency. In the numerator, you’d have the number of groups of three scarves considering arrangement, while in the denominator, it would still be without it. Thus, multiplying by 3!/2! does not make sense.

Hope that clears things up!

Bhavita. · Answer

Bunuel

Is it like until unless its not mentioned as identical explicitly in the question stem we are supposed to treat it like different objects. Also what if it were identical how would the solution be like (I am not able to solve considering them as identical any help would be helpful)

Bunuel · Answer

The scarves are real, physical items. Even if two are the same color, they are still distinct objects. We cannot treat them as identical.

sunshineeee · Answer

I like the solution - it’s helpful.

Karym · Answer

I like the solution - it’s helpful.

Karym · Answer

I like the solution - it’s helpful.

At least one grey scarf	Exactly two grey scarves
		\(\frac{1}{720}\)
		\(\frac{7}{720}\)
		\(\frac{7}{40}\)
		\(\frac{7}{24}\)
		\(\frac{1}{2}\)
		\(\frac{17}{24}\)

At least one grey scarf	Exactly two grey scarves
		\(\frac{1}{720}\)
		\(\frac{7}{720}\)
		\(\frac{7}{40}\)
		\(\frac{7}{24}\)
		\(\frac{1}{2}\)
		\(\frac{17}{24}\)

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