Try, Try Again

•What does probability mean?

•What does a probability of zero mean?

•What about a probability of 1.0?

Consider the following.

•What’s the probability of snow tomorrow?

•What’s the probability of the brown phone ringing during class today?

•What’s the probability that Kobe will sink a free-throw?

Suppose we know that Kobe made his first free-throw shot. What’s the probability that he sinks

two in a row? Is that higher, lower, or the

same?

• If you flip a coin, what’s the probability of “heads”? How do you know?

• If you roll a die, what’s the probability of getting a three? Again, how do you know?

• If you roll a die, what’s the probability that you get an odd number? How did you get that?

• If you roll a die 60 times, how many threes do you expect to get?

• Suppose you have a paper cup and flip it in the air. What’s the probability that it lands on its side? What could you do to figure it out?

• Suppose you have a fair coin and flip it twice. What’s the probability that you get two heads?

A probability is a number between zero and one (inclusive), so you can represent it as a fraction, a

decimal, or a percent. Which should we use? At this stage, it really doesn’t matter. But, consider the

following.

• For theoretical probabilities using dice, fractions are exact while the others are approximations. If a we always says that the probability of rolling a three is “point one six seven,” we might be missing the elegance of “one sixth.”

• If we use fractions, we may have developed the reflex that we have to express every answer in lowest form. Not true! When we do two-dice sums, for example, the probability of rolling a 5 is 4/36. This is better than 1/9 because it contains useful information: there are four ways to get a five among the 36 possibilities.

• For empirical probabilities—which are approximations by nature — decimals and percents make it easier to compare.

The Task

•The task is to explain why Aloysius is wrong—using two different approaches.

•The first is theoretical: make a diagram

•or systematic list.

•The second is empirical: simulate Steve going to the free-throw line many times.

Things to Consider

•What is the probability that he makes his first shot?

•Now ignore the first shot. What’s the probability he makes the second?

For the “theoretical” part, you might ask the

following:• What are the different possible things that can happen when

Steve shoots twice?

• What are the different possibilities for his first shot? What are their probabilities?

• Suppose Steve took these two shots 100 times. Of those 100 pairs of shots, how many times will he make his first shot? And out of this, how many times will he make his second shot? So what’s the chance that he makes the first shot and makes the second shot?

• Be sure to explain what Aloysius was thinking. Why is his approach wrong?

For the “empirical” part, you might ask the

following:•What do you suppose it means to simulate his shots? (Important: To solve this problem, do not crumple up a piece of paper and try to pitch it into the wastebasket. Why not? Because you can’t guarantee that your shot will hit an average half the time.)

•How could you simulate his first shot? How could you make the probability ½?

•How are you recording your data?

•How many trials do you think you need?

Poster Problems - Try, Try Again Slide #1

✔︎ ✔︎ ✔︎ ✔︎

P(HH) ~ 4/18 or about 0.22

Poster Problems - Try, Try Again Slide #2

Why were the empirical results different from the

theoretical results?

How could we get empirical results that are closer to the theoretical results?

Combining DataTotal Simulations Total Favorable

Outcomes

P = 1/4

H MH HM HM M

2 POINTS

hit miss

hit

2points

1point

miss

1point

0points

FIRST SHOT

SEC

ON

D

SH

OT

P = 1/4

hit

miss

H+HH+MM+H

M+M

︎ ✔︎

Theoretical Approach: Make a systematic list, table, or diagram

Empirical Approach: Try it many times simulating the thing you’re trying to figure out

Theoretical

Poster Problems - Try, Try Again Slide #3

The Task

•Penelope works in a group with Quentin, Rochelle, and Sid.

•The teacher always chooses a group member at random to present the group’s solution to the class.

•On this day, Penelope’s group has to present twice.

•What’s the probability that the same student gets chosen both times?

Consider

•What is the probability that Penelope gets picked for the first solution?

•How would you simulate picking one student?

After Task Discussion

•Why do empirical answers vary?

•Do empirical results center on theoretical results?

•There are several ways to show the theoretical result. Which do you prefer?

•There are several ways to do the simulation. Which do you like?

neighbors not neighbors

Poster Problems - Try, Try Again Slide #4

If you roll two dice, what’s the chance that the numbers you get will be “neighbors”—that is, that they differ by one? For this task, use a theoretical approach.

Homework