Dora and Her Game of Go Fish

Dora wanted to play a good game of Go Fish with her friends. So She decided to invite 5 of her other friends over: Boots the Monkey, Benny the Bull, Señor Toucan, Liza the Lizard, and her cousin Diego.

Dora: Hey guys, lets play on top of that stump over there.

Boots: I want to sit with you Dora!

Benny: Me too!

Diego: Dora might cheat, I won't sit beside her.

a) Oh no Dora and her friends have a problem. How many ways can Dora and her friends sit at the tree stump to play cards when Dora has to sit with Boots and Benny? In addition Diego doesn't want to sit with Dora.

So to first solve this problem, let's analyze this question a bit more. Since Dora must sit with Boots and Benny, how many ways can they sit together? They are consider a single object in this group which means that there are only 4 people that has to be arranged at the tree stump. We can then find the number of ways that those four people can sit at the round stump. Remember that Diego doesn't want to sit with Dora...

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b)
Dora: Okay, we've sorted this out. Now let's play some go fish.

Liza: I don't know how to play.

Benny: I do. In go fish, we all first have five cards, and the objective of this game is to get rid of our hands by making pairs. We can get pairs by drawing a card or asking if another player has the card you need. First one to lose all of the cards in their hand wins!

Liza: Wow that sound's easy enough, you are so smart Benny.

Benny: Why thank you...

Diego: Let's just play already. I'll deal.

So Diego give each player five cards. Boots got his 5 cards first. Him being a gambler and all wants to have a little fun.

Boots: Okay I'm going first and I bet I can win this by just picking up only one card.

Liza: Let's see you try Boots!.

Woah, Boots is being a bit gutsy. What is he going to do? What is the probability of Boots winning by drawing only one card if: i) No one has the card he needs ii) One person has the card he needs iii) Two people have the card he needs?

In order to solve this we have to find the probability of getting a 5 card Poker hand of 2 Pairs or a 4 of a Kind. Then we must solve the probability of getting that card that Boots needs to win.

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Boots: Woot I won in one draw. I told you I could win. Hahahaha!

Diego: Darn you Boots!!

Dora: Thank you guys for helping me with today's problems. What was you favorite part of today?...(silence)...me too!!!