Randomness, Probability, and Simulation ACT Info

Investigating Randomness Pretend that you are flipping a fair coin. Without actually flipping a coin, imagine the first toss. Write down the result you see in your mind, heads (H) or tails (T). Imagine a second flip. Write down the result. Keep doing this until you have 50 H’s or T’s written down. Write your results in groups of 5 to make it easier to read, like this: HTHTH, TTHHT, etc

A run is a repetition of the same result A run is a repetition of the same result. In the example in #3, there is a run of two tails followed by a run of two heads in the first 10 coin flips. Read through your 50 imagined coin flips, and count the number of runs of size 2,3,4,etc. Record the number of runs in a table like this: Run Length 2 3 4 5 6 7 8 Frequency

6) Get data from one other person (write below your data in the chart) 7) Use your calculator to generate a similar list of 50 coin flips. Let 1 represent heads and 0 represent tails. (use randInt feature) 8) Record the number of runs the same as before 9) Compare the three results. Did you, your classmates’ data, or your calculator have the longest run? How much longer? 10) Meet with not your partner: How is the data distributed (yours vs. calculator)

Short Run Regularity Myth Which looks more probable? HTHTTH TTTHHH If a basketball player makes several consecutive shots, will he/she make the next shot? Players perform consistently, not in streaks. Runs are more common than our intuition expects

Law of Averages Myth If you toss a coin six times and get TTTTTT, what do you think will be the next toss result? If I have become a mother of 5 girls, will I have a boy for the 6th? Myth is that future outcomes must make up for an imbalance like 6 straight tails. In the long run, tails will appear half the time Coins, dice have no memories. They are not trying to make things even out. After 10,000 tosses, the results of the first six are negligible.

Shaquille O’Neal: Simulation In 2000, Shaq won all three MVP awards. But he was never a “complete” player. He was a 50% free- throw shooter. Let’s assume that every time Shaq steps up to the free-throw line, the probability that he will make the shot is 0.5. We want to know how likely he is to make at least 3 free throws in a row out of 10 attempts (a “run” of 3 or more)

Shaq: Simulation State assumptions Assign digits to represent outcomes Simulate many repetitions (40) What is your estimate of the probability Combine your data with 2 other people, what is the probability now? p= 0.5, free throws are independent Digits 0-4= made shot, digits 5-9: missed Generate 10 digits at a time, count if he had 3 or more runs

Independence Two random phenomena are independent if knowing the outcome of one does not change the probabilities for the outcomes of the other.

Rock-Paper-Scissors What is the probability that a game will result in a winner? State assumptions Assign digits to represent outcomes Simulate 30 repetitions. What is your estimate of the probability?

First Ace Begin with a standard deck of cards. Shuffle and draw a card. Replace the card and shuffle, draw again. Continue until you draw an ace, or until you draw 10 cards (whichever comes first). What is the probability of drawing an ace in 10 draws? State assumptions Assign digits Simulate 20 repetitions What is your estimate of the probability? Use calculator APPS prob sim