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Probability Coins Station 1 Probability is defined as the chances that something will happen. It is based on events being random, or equally likely. Probability can be written as a fraction. This shows how many times the event should occur out of the total options available. Fractions can be turned into percents by dividing the top number by the bottom number and multiplying by 100. Probabilities are mathematical predictions and although they are accurate based on randomness, no one can ever predict the outcome of a random event with certainty. The more times an event happens, the closer the total amount will be to the predicted amounts. For example if you flip a coin the chance of getting heads is 1/2 because when you flip a coin there are two possible outcomes (heads or tails) and heads is just one of those the possibilities. Therefore the fraction: The number of correct choices = 1 way to get heads The total number of correct and incorrect choices possible = 2 sides (heads or tails) You can turn this fraction into a percent chance of getting heads by dividing the fraction (1 divided by 2 = 0.5) and then multiplying by 100 (0.5 x 100 = 50). Therefore the chances of getting heads when you flip a coin is 50%. 1. This is a prediction. Lets test this prediction. 2. Using the coin at your desk, flip it 2 times. Put a tally mark for each event. Trial Predicted Actual Heads Actual Tails Tails = 1 Total A Heads = 1 B Heads = ___ Tails = ___ 10 C Heads = ___ Tails = ___ 30 3. Start again but now flip 10 times. Fill in the predicted and actual numbers. 4. Do it one more time with 30 flips. 5. Which trial was closest to the predicted outcome? Explain your answer. 6. What are some factors that might make the coin flip NOT a random event? 2 Probability Dice Station 2--Refer to your station 1 answer sheet to help you with this station. 1a. Examine the black die on your table. 2b. Suppose you wanted to roll the number 6. 3c. What are the chances of rolling a 6 on one die expressed in a fraction? The number of correct choices The total possible of correct and incorrect choices 4d. What are the chances, expressed as a percent, of rolling a six on one die? Round to the nearest whole number, show your work. 1b. Examine the red die on your table. 2b. Suppose you wanted to roll the number 6. 3b. What are the chances of rolling a 6 on one die expressed in a fraction? The number of correct choices The total possible of correct and incorrect choices 4b. What are the chances, expressed as a percent, of rolling a six on one die? Round to the nearest whole number, show your work. 1c. Examine the blue die on your table. 2c. Suppose you wanted to roll the number 6. 3c. What are the chances of rolling a 6 on one die expressed in a fraction? The number of correct choices The total possible of correct and incorrect choices 4c. What are the chances, expressed as a percent, of rolling a six on one die? Round to the nearest whole number, show your work. 1d. Examine the green die on your table. 2d. Suppose you wanted to roll the number 6. 3d. What are the chances of rolling a 6 on one die expressed in a fraction? The number of correct choices The total possible of correct and incorrect choices 4d. What are the chances, expressed as a percent, of rolling a six on one die? Round to the nearest whole number, show your work. 1e. Examine the yellow die on your table. 2e. Suppose you wanted to roll the number 6. 3e. What are the chances of rolling a 6 on one die expressed in a fraction? The number of correct choices The total possible of correct and incorrect choices 4e. What are the chances, expressed as a percent, of rolling a six on one die? Round to the nearest whole number, show your work. 5a. Have each person at the table choose a die. Roll the dice over and over at the same time until someone at your table rolls a 6. 5b. Cirlce the color dice rolled at your table:! black! red blue green yellow 5c. Out of the colors your group is rolling, which one do you predict will roll a 6 first? Explain your answer. 5d. Which color die rolled a 6 first? What are your thoughts about this? Probability Cards Station 3--Refer to your station 1 answer sheet to help you with this station. 1. Examine your deck of cards. Count and record the total number for the following: Total cards = _____________ Spades = _____________ Hearts = _____________ Diamonds = _____________ Each number (5w, 8s, 10s, etc) = ______________ Red cards = _____________ Black cards = _____________ 2. Determine the probability, in both fraction and percent form for drawing one card out of a full deck that is: Event Fraction Percent a black card an ace a red king a number under 10 (not including ace) any face card (J, Q, K) an even number card that is also black 3. Test one of those probabilities above by drawing out of the full deck as many times as it takes to get what you are looking for. Record and compare the predicted and actual outcomes. Probability Coins Station 4: In stations 4-6 you will use what you learned from stations 1-3 and apply it to calculating the probability of multiple events. So far we have only calculated the chances of single events. You can use math to predict the outcome of multiple events too. To do this simply multiply the chance of the first occurence times the chance of susequent occurences to determine the chance of a SERIES of events. For example; What if you want to know the chance of flipping a coin two times in a row and getting heads BOTH TIMES. Simply multiply the independent events and you have your answer. Event 1 = HEADS ! ! TIMES ! ! X!! ! ! ! =! ! ! ! Event 2 = HEADS Therefore the chances of flipping one heads followed by ANOTHER flip of heads is 1/4 or 25%. 1. Fill in the table below in both fraction and percent. Trial Chain of events A H, H B H, T, T, H, T C T, T, H, T, T, H, T, H fraction percent 2. Test how many times it takes you to achieve the order of the event in trial B. Remember to start over everytime you fail and count this as a new attempt. 3. How many times did it take? 4. How does this compare to the predicted? Probability Dice Station 5: In stations 4-6 you will use what you learned from stations 1-3 and apply it to calculating the probability of multiple events. 1. What is the probability of rolling: Trial Chain of events A 2, 4 B 1, 5, 2 C 6, 6, 6, 6, 6 fraction percent 2. Look back at your answer sheet for station 4, trial C. Compare and contrast trial C from station 4 and from station 5. 3. There is another rule in probability that is sometimes stated “chance has no memory”. In other words, when you roll a die, the previous rolls HAVE NO EFFECT on the future rolls. In other words, if you take a 12-sided die and roll it once and get a 12, the chances for rolling a 12 the next time you roll it are STILL THE SAME, 1/12 or about 8%. It doesn’t seem logical, but this is how probability works. Let’s take some data using the 12-sided die and see what happens. 4. Roll the 12-sided die 36 times. Chance should say each number should come up three times....but this is just a prediction and rarely, if ever is perfect. Use tally marks to record the ACTUAL numbers rolled. 1 2 3 4 5 6 7 8 9 10 11 12 5. Were there any numbers never rolled? Which ones? 6. Which ones seemed to be the “lucky” numbers? 7. If you had to guess which number would be next, what would you say? Does the table about what has happened before influence your answer? Explain. Probability Cards Station 6: In stations 4-6 you will use what you learned from stations 1-3 and apply it to calculating the probability of multiple events. Imagine two Texas holdem poker hands: Hand 1! ! K of clubs! ! Q of hearts! ! ! ! ! Hand 2! ! 7 of hearts! ! 10 of hearts ! ! Flop Ace of hearts 3 of clubs 9 of spades Hand one is hoping for a straight...two more cards will be drawn. They need one card to be a Jack of any suit and one to be a ten of every suit. What are their chances? Show your work. (hint: the deck HAD 52 cards to start but now has only 45 left) (hint: calculate the first card being a jack and the second being a ten, then ADD this to the first card being a ten and the second card being a jack) Hand two is hoping for a flush and needs two more hearts to be drawn. Remember the deck only has 45 cards to start the drawing of two more cards. Show your work. If neither hand gets what they are looking for Hand 1 has a higher probability of winning because of its high cards. Who should bet it all? Explain