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**Making Predictions with Theoretical Probability**

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Warm Up In a standard 52-card deck, half of the cards are red and half are black. The 52 cards are divided evenly into 4 suits: spades, hearts, diamonds, and clubs. Each suit has three face cards (jack, queen, king), and an ace. Each suit also has 9 cards numbered from 2 to 10. Dawn draws 1 card, replaces it, and draws another card. Is it more likely that she draws 2 red cards or 2 face cards? Luis draws 1 card from a deck, 39 times. Predict how many times he draws an ace. Suppose a solitaire player has played 1,000 games. Predict how many times the player turned over a red card as the first card.

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**You can make quantitative predictions based on theoretical probability.**

You can use theoretical probabilities to make qualitative predictions or compare how likely events are. There is a connection between relative frequency and probability. If you know the structure of the generating mechanism of the probability experiment, you can approximate the relative frequency of a series of random selections in a quantitative or qualitative way using theoretical probability. You can use mathematical models to generate the possible non-overlapping outcomes of the probability experiment and find the probabilities of the model.

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To make predictions using theoretical probability use the theoretical probability to write an equation or use a proportion to make the prediction for a given experiment.

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Create a tree diagram to show the outcomes and sample space for flipping a coin and rolling a number cube. Use the diagram to find the probability of landing on heads and getting a 6.

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**A bag contains same-size game pieces**

A bag contains same-size game pieces. There are 6 purple pieces, 8 blue pieces, and 5 green game pieces after Geri removed one purple game piece without looking. Find the theoretical probability that the next game piece Geri picks is the same color and the probability that it is a different color. Then explain whether it is likely that the next game piece she picks will match.

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**A spinner has eight equal sections numbered from 1–8**

A spinner has eight equal sections numbered from 1–8. Predict how many times the pointer will land on a multiple of 3 in 300 spins.

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**Blake volunteers at the animal shelter**

Blake volunteers at the animal shelter. He has an equally likely chance of being assigned to the dog, cat, bird, or reptile section. If he volunteers 30 times, about how many times should he expect to be assigned to the cat or reptile section?

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**Lea pulls a blue tile out of a bag of same-size tiles without looking**

Lea pulls a blue tile out of a bag of same-size tiles without looking. There are 4 purple tiles, 6 blue tiles, and 8 green tiles left in the bag. Find the theoretical probability that the next tile she picks is the same color and the probability that it is a different color. Is it likely the next tile she picks will match?

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A company randomly assigns employees four-digit security codes using the numbers 1 through 4 to activate their accounts. Any of the digits can be repeated. Is it likely that more than 3 of the 1,280 employees will be assigned the code 4113?

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**A bag contains 6 red marbles, 2 white marbles, and 1 gray marble**

A bag contains 6 red marbles, 2 white marbles, and 1 gray marble. You randomly pick out a marble, record its color, and put it back in the bag. You repeat this process 45 times. How many white or gray marbles do you expect to get?

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**Horace is going to roll a standard number cube and flip a coin**

Horace is going to roll a standard number cube and flip a coin. He wonders if it is more likely that he rolls a 5 and the coin lands on heads, or that he rolls a 5 or the coin lands on heads. Which event do you think is more likely to happen? Find the probability of both events to justify or reject your initial prediction.

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Exit Ticket You roll a six-sided number cube 270 times. Predict how many times you will roll a number less than 3. The equal sections of a spinner have one of the following letters: A, B, A, C, B. Predict how many times the pointer will land on a consonant in 60 spins. In Kim’s homeroom, 4 students have no siblings, 9 students have 1 sibling, 8 students have 2 siblings, and 2 students have 3 or more siblings. Is it likely that a student chosen at random will have 2 or more siblings? Explain. Drew rolls two number cubes 75 times. Is it likely that she will get a sum of 3 more than 5 times? Explain.

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