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The following cards are dealt to three people at random, so that everyone gets the same number of cards. What is the probability that everyone gets a red card?

 

[There are three red cards and three blue cards]

 Dec 27, 2022
 #1
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Just to start off, I want you to read my answer and explanation, and to make sure you do that, I will be solving a SIMILAR question, but not exactly. Here is the question I will be solving: 

 

The following cards are dealt to FOUR people at random, so that everyone gets the same number of cards. What is the probability that everyone gets a red card?

 

[There are FOUR red cards and FOUR blue cards]

 

To start off, if we are going to calculate probability, that means that there is going to be a fraction in the form of:

successes/total

 

Let's find the total first. To start, we have 8 cards for the first person, 7 cards for the second, 6 cards for the third and 5 for the fourth, multiplying gives us:

 

8*7*6*5 total possibilities (we aren't going to calculate this just yet)

 

Then, we need to find the probability that everyone gets a red card. There are only 4 cards, and we can put them respectively to each of them 4! different ways (4 * 3 * 2 * 1).  Therefore the probability is:

 

4!/(8*7*6*5)

This simplifies to:

3/(7*6*5)

This simplifies to:

1/(7*2*5)

Which is 

1/70

 

Note: This is not the answer to your question, but the answer to a similar question. I advise you to read this explanation, and if you find any errors, please tell me (as counting isn't my strong subject, there might be something wrong).

 Dec 27, 2022
 #2
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Let's imagine that the marbles are drawn one by one. Person 1 draws their first marble then their second marble. Then, person 2 draws their first marble then their second marble. Finally, person 3 draws their first marble then their second marble.


That gives us 6 separate draws, of which 3 are red. That means that there are  (6 C 3)  ways for 3 red balls to be drawn overall.


How many ways are there for every child to get a red marble? The number of red marbles is the same as the number of children, so each gets exactly 1 red marble.


For person 1, that could be their first marble or their second marble, so there are 2 ways for person 1 to have 1 red marble. That is true for person 2 and person 3 as well. Therefore, the number of ways for everyone to have a red marble is  2⋅2⋅2  or  2^3 .


That gives the following probability:


2^3 / (6 C 3) = 2 / 5

 Dec 28, 2022
 #3
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Note: I used marbles and children instead of "cards and people". But the answer would be exactly the same.

Guest Dec 28, 2022

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