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]
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:
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:
This simplifies to:
This simplifies to:
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).
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