Algebra 1 Summer Institute 2014

Unit5 – Probability

Essential Question

How can one generate a random sample via simulation?
What do the results of a computer random number generator mean?
What does “the probability of tossing a head equals one-half” mean?

Learning Goals

Explain whether a given sampling technique is random
Use random sampling to draw an inference or answer a question
Explain the probability of events from a discrete probability distribution
Explain the probability of events from a continuous probability distribution
Determine the probability distribution in a model context
Generate an estimate for the probability distribution using technology.
Identify the sample space for a given context.
Design a simulation for compound events.

Associated MAFS

MAFS. 7.SP.3: Investigate chance processes and develop, use, and evaluate probability models

3.5: Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around ½ indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

3.6: Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.

3.7: Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.

Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected a random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.
Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper clip will land open end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies.

3.8: Find the probabilities of compound events using organized lists, tables, trees, and simulation.

Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
Represent sample spaces for the compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g.,”rolling double sixes”), identify the outcomes in the sample space which compose the event.
Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?

MAFS.912.S-IC.1: Understand and evaluate random processes underlying statistical experiments

1.1: Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

1.2: Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says that a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?

MAFS.912.S-ID.2: Summarize, represent, and interpret data on two categorical and quantitative variables

2.5: Summarize categorical data for two categories in two-way frequency table. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

Student Misconceptions:

1.The Binomial Distribution and N

The Misconception:

If a fair coin is flipped n times (with n being an even number), the potential result of “equality” (i.e., getting as many heads as tails) is more likely if n is large rather than small.

Undoing the Misconception:

Show through example. If you flip a fair coin an even number of times, what is the probability of getting as many heads as tails? Note that as the number of coin flips, N, increases, the probability of observing “equality” decreases. If you flip that same fair coin an even number of times, the probability of getting a result that approximates the expected value (Np ) goes up as N increases.

2.A Random Walk With a Perfectly Fair Coin

The Misconception:

If a perfectly fair coin is flipped 50 times with you betting that each flip’s outcome will be heads while a friend bets against you, then your ongoing cumulative performance--based on $1 given by the loser to the winner after each flip--will cause you to be “in the black” (i.e., with positive earnings) about as often as you are “in the red” (i.e., in debt to your friend) across the series of coin flips.

Undoing the Misconception:

Show through example. An area of statistics dealing with random walks has been used to investigate what happens when a sum is calculated, over time, for a random process.

3.Two Goats and a New Car

The Misconception:

If you are shown three curtains and told (a) that a new car is parked behind one of the curtains while a goat sits behind each of the other curtains and (b) to choose a curtain for your prize, it doesn’t matter whether you do or don’t “switch” your selection after one of the two curtains you didn't select is opened and reveals a goat.

Undoing the Misconception:

Show through example. Activity will be examined in the Institute.

4.Identical Birthdays

The Misconception:

In a small, randomly selected group people, it’s unlikely that two or more of the individuals have the same birthday.

Undoing the Misconception:

Show through example.Activity will be examined in the Institute.

Activity progression:

Activity 2: The Fair Unfair Polarization

Activity 1: The EP Verification

Activity 3: The Poker Manipulation

Video: Did you Know?

Activity 4: The Birthday Paradox

Activity 5: The Stick, Flip, and Switch Corollary