Complex(er) Probability

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There are three different sandwiches on a shelf. There are: 4 egg sandwiches, 5 cheese sandwiches, 2 ham sandwiches. Erin takes 2 of these sandwiches. What is the probability that she will get 2 different types of sandwiches?
 
There are three different sandwiches on a shelf. There are: 4 egg sandwiches, 5 cheese sandwiches, 2 ham sandwiches. Erin takes 2 of these sandwiches. What is the probability that she will get 2 different types of sandwiches?
There are a total of 11 sandwiches and four of them are egg sandwiches so the probability she takes an egg sandwich is 4/11. There are then 10 sandwiches and 3 of them are egg sandwiches so the probability she takes a second egg sand wich is 3/10. The probability both sandwiches are egg sandwiches is (4/11)(3/10)= 6/55.

Now, what is the probability both sandwiches are cheese sandwiches? What is the probability both sandwiches are ham sandwiches? The sum of those is the probability she gets two sandwiches are the same. And one minus that is the probability she gets two different sandwiches.


(This really should say "Erin takes 2 of thes sandwiches at random". If it were me, the probability of get 2 different types of sandwiches would be 0. I don't like cheese or ham sandwiches!)
 
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