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Bunuel

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19 Nov 2024, 01:16

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Vineyashdev

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.

Are each grey scarves identical? same goes for all colours.
If yes how should I have interpreted this information ?

There are three different grey scarves, two different yellow scarves, and so on. I think imagining a real-life situation without overthinking it might help.

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10 Jan 2025, 03:30

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Bunuel

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: \(P(at \ least \ one \ grey \ scarf) = 1 - \frac{7}{10} \times \frac{6}{9} \times \frac{5}{8} = \frac{17}{24}\).

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: \(P(exactly \ two \ grey \ scarves) = \frac{C^2_3 \times C^1_7}{C^3_{10}}= \frac{3 \times 7}{120} = \frac{7}{40}\).

Correct answer:

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

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

Could you pls tell me where I am going wrong

1. At least 1 Grey Scarf

GOO+GGO+GGG (O-other colours)
3/10*7/9*6/8 + 3/10*2/9*7/8+3/10*2/9*1*8 = 174/720

2. Exactly 2 Grey
3/10*2/9*7/8 = 7/120

I know the answers I have got are nowhere close to the options given, but I want to know where I am going wrong and clear my concepts.
Thank You

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Bunuel

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13 Jan 2025, 01:48

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MURALIKA

Bunuel

That's a common mistake when solving with the probability approach. The key is that you need to account for different scenarios in which each case can occur.

For example, picking exactly two grey scarves (the GGX scenario) can happen in three different ways: GGX, GXG, or XGG. Since each has a probability of 7/120, the total probability becomes 7/120 * 3 = 7/40.

For at least one grey scarf, we'd have 3/10 * 7/9 * 6/8 * 3!/2! + 3/10 * 2/9 * 7/8 * 3!/2! + 3/10 * 2/9 * 1/8 = 17/24.

For the GXX case, we multiply by 3!/2! because Grey, Not Grey, Not Grey can occur in 3!/2! = 3 ways: GXX, XGX, or XXG, which accounts for the different permutations of G, X, and X.

Similarly, for the GGX case, we also multiply by 3!/2! = 3 because GGX, GXG, or XGG are distinct scenarios.

The third case, GGG, can only occur in one way, so no factorial correction is applied to it.

Hope it's clear.

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