Lesson 10 Slacker’s Simulation Solidify Understanding

Jump Start

Find the following:

1.

The probability that a fair coin is tossed twice and lands on heads both times.

2.

The probability that a spinner with $6$ colors (red, blue, green, yellow, pink, and orange) lands on red, and then a coin is flipped and it lands on tails.

Learning Focus

Perform a simulation to determine if an event can occur.

Is there a way to test a claim without performing a study on actual subjects?

Why don’t real results always match theoretical probabilities?

Open Up the Math: Launch, Explore, Discuss

I know a student who forgot about the upcoming history test and did not study at all. To protect his identity, I’ll just call him Slacker. When I reminded Slacker that we had a test in the next class, he said that he wasn’t worried because the test has $10$ true/false questions. Slacker said that he would totally guess on every question, and since he’s always lucky, he thinks he will get at least $8$ out of $10$ . That’s what he did on the last quiz and it worked great.

I’m skeptical, but Slacker said, “Hey, sometimes you flip a coin and it seems like you just keep getting heads. You may only have a $50 / 50$ chance of getting heads, but you still might get heads several times in a row. I think this is just about the same thing. I could get lucky.”

1.

What do you think of Slacker’s claim? Is it possible for him to get $8$ out of $10$ questions right? Explain.

I thought about it for a minute and said, “Slacker, I think you’re on to something. I’m not sure that you will get $80 %$ on the test, but I agree that the situation is just like a coin flip. It’s either one way or the other and they are both equally likely if you’re just guessing.” My idea is to use a coin flip to simulate the T/F test situation. We can try it many times and see how often we get $8$ out of $10$ questions right. I’m going to say that if the coin lands on heads, then you guessed the problem correctly. If it lands on tails, then you got it wrong.

2.

Try it a few times yourself. To save a little time, just flip $10$ coins at once and count up the number of heads for each test.

	# Correct (Heads)	# Incorrect (Tails)	% Correct
Test 1
Test 2
Test 3
Test 4
Test 5

Did you get $8$ out of $10$ correct in any of your trials?

3.

Based on your trials, do you think Slacker is likely to get $80 %$ correct?

4.

Check out the histogram that represents the data from the whole class. Now what do you think of Slacker’s chances of getting $80 %$ correct? Explain why.

Pause and Reflect

5.

What would you expect the graph to look like if you continued to collect samples? Why?

6.

Based upon your understanding of this distribution, what would you estimate the likelihood of Slacker getting $80 %$ on the test without studying?

Ready for More?

Padma has been playing a board game with her friends where the moves are determined by the sum of the numbers on three number cubes. In the last game, one of Padma’s friends needed to roll a sum of $5$ on the cubes to win and it took her a lot of turns to get it. One of the players remarked, “I don’t know why it took so long, since $5$ is no more or less likely than any of the other numbers between $3$ and $18$ .” Padma decided to investigate this claim with a simulation. She rolled $3$ number cubes $100$ times and got this distribution:

a.

Which of these statements are true about rolling $5$ in this game?

A.

It is not possible to roll 5 in this game.

B.

It is not very likely to roll $5$ in this game.

C.

Rolling $5$ is less likely than rolling $4$ in this game.

D.

Rolling $5$ is less likely than rolling $10$ .

E.

Rolling $5$ and rolling $16$ have the same likelihood.

b.

Explain each of your choices.

Takeaways

Simulation:

Law of Large Numbers:

Vocabulary

simulation
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we used simulation to model the outcome of a random event. Using many trials, we created a distribution and used it to predict the likelihood of an event.

Retrieval

$A = [\begin{matrix} 1 & 0 \\ - 1 & 2 \end{matrix}]$ and $B = [\begin{matrix} 5 & - 1 \\ 2 & - 3 \end{matrix}]$

1.

Find $A + B$ and $B + A$ . Are they the same?

2.

Find $A B$ and $B A$ . Are they the same?

3.

In a group of $100$ students, $40$ are taking algebra, $30$ are taking biology, and $20$ are taking both algebra and biology.

a.

Draw a Venn diagram to represent this information.

b.

If a student, chosen at random, is taking algebra, what is the probability that he or she is taking biology? (Let $A$ be algebra and $B$ biology.)

c.

Which notation means the same thing as the question in part b?

A.

$P (A)$

B.

$P (A \cup B)$

C.

$P (A \cap B)$

D.

$P (B | A)$

E.

$P (A | B)$