Counting and Probability We’ll change seats on Thursday

Quiz #5: According to the Census Bureau, the number of people X in a randomly selected U.S. household follows the probability distribution given in the table below. Sketch a histogram that displays the probability distribution of X Calculate P(X>2) Interpret P(X >2) Find the expected value of X Explain what the expected value tells you. Number of People (X) 1 2 3 4 5 6 7 Proability (p) 0.25 0.32 0.17 0.15 0.07 0.03 0.01

Deal or No Deal http://www.agame.com/game/deal-or-no-deal-iwin

Deal or No Deal What’s the probability that you choose the $1,000,000 case to begin with? If you played the game many many times, what is the expected value of the amount that your case contains? Suppose you get lucky and pick the $1,000,000 case to begin with (you don’t know this of course). How likely do you think you would be to keep turning down the banker’s offers and to actually leave the show with $1,000,000?

Deal or No Deal Record which cases we open in each round and the banker’s offer. Decline all of the offers until we get to see what’s in the case. Round Cases opened Banker’s Offer Average $ of remaining cases Banker’s offer compared to average $

New Seats! One partner grab a computer again. We’ll meet back up with our old partners to finish the Deal or No Deal work. (If you finished, just meet up to make sure you have the data) New partners- we will do another round (modified slightly)

Round Cases opened Banker’s Offer Average $ of remaining cases Banker’s offer compared to average $

Deal or No Deal Leave your computer at the table, find your new seat Play the game 4 times. Try to finish with as much money as you can. Record data in the same way as before (but you can choose when you accept the offer) Did we make a good deal?

Ice Cream Sundaes The ‘28 Choices’ ice cream shop offers 28 flavors of ice cream, 11 different toppings, and 3 sizes of bowl for their ice cream sundaes. In how many ways can you order a sundae with one scoop of ice cream and one topping? How many ways can you order a one-scoop sundae?

Assigned seating I decide to randomly assign seats for 25 students in one class. How many different seating assignments are possible?

Factorial 5! 12! N!

Permutations Order matters nPk = n!/ (n-k)!

Combinations Order does not matter nCk= n!/k!(n-k)!

Ice Cream (again) You decide to stop by ‘28 Choices’ for some ice cream. With 28 flavors to choose from, how many different ways are there to order: A) a cone with three scoops if you care about the order of flavors B) a cone with three scoops of different flavors if you care about the order of flavors? C) a cone with three scoops of different flavors (order doesn’t matter)? D) a cone with three scoops of ice cream (order doesn’t matter)?

Pick 3 lottery Simple lottery wager, the “Straight” from the Pick 3 game of the Tri- State Daily Numbers offered by New Hampshire, Maine, and Vermont. You pay $1 and choose a three-digit number. The state chooses a three-digit winning number at random and pays you $500 if your number is chosen. What’s the probability that you win? What is the long-run average? 10*10*10= 1000 possible winning numbers. P(you get winning)= 1/1000, P(didn’t win)= 999/1000 Value: $0 $500 P(v): 999/1000 1/1000

Looking for aces Suppose we deal two cards from a well-shuffled, standard deck. What’s the probability that both are aces? Method 1-Tree diagram Method 2- Permutations, Card 1 and Card 2 52 possibilities for Card1, 51 for Card2 or 52P2. Aces have 4P2 so, P(two aces)=#of arrangements with two aces/ # of arrangements of first 2 cards = 4P2/52P2 Method 3: Combinations, we really don’t care about the order so could use these too

Looking for aces Suppose we deal two cards from a well-shuffled, standard deck. What’s the probability that both are aces? Method 1-Tree diagram Method 2- Permutations, Card 1 and Card 2 52 possibilities for Card1, 51 for Card2 or 52P2. Aces have 4P2 so, P(two aces)=#of arrangements with two aces/ # of arrangements of first 2 cards = 4P2/52P2 Method 3: Combinations, we really don’t care about the order so could use these too

Pick 4 In the Pick 4 game of the Tri-Stat Daily Number, you pay $1 and choose a four-digit number. The state chooses a four-digit winning number at random and pays you $5000 if your number is chosen. A) What’s the probability that you win? B) Find the expected value of your winnings.

Quiz #6. State if a permutation or a combination and then find the number of possibilities

Texas Hold ‘Em In Texas Hold ‘Em, players are initially dealt two cards. One very strong starting hand is called “big slick,” which consists of an ace and a king (not necessarily of the same suit). If you deal two cards from a well-shuffled deck, what’s the probability of getting “big slick”?

Pascal’s Triangle- Connection to combination

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