Theoretical vs Experimental Probability

Experimental probability: Probability based on a collection of data. Will have a table of results or data from the experiment(s)!

What is the difference between theoretical probability and experimental probability? Theoretical probability shows what should happen in an experiment. Experimental probability shows what actually happened.

Computing theoretical probabilities:counting methods Great for gambling! Fun to compute! If outcomes are equally likely to occur… Note: these are called “counting methods” because we have to count the number of ways A can occur and the number of total possible outcomes.

Counting methods: Example 1 Example 1: You draw one card from a deck of cards. What’s the probability that you draw an ace?

Counting methods: Example 2 Example 2. What’s the probability that you draw 2 aces when you draw two cards from the deck? This is a “joint probability”—we’ll get back to this on Wednesday

Counting methods: Example 2 Two counting method ways to calculate this: 1. Consider order: Numerator: AA, AA, AA, AA, AA, AA, AA, AA, AA, AA, AA, or AA = 12 .   52 cards 51 cards Denominator = 52x51 = 2652 -- why?

Counting methods: Example 2 2. Ignore order: Numerator: AA, AA, AA, AA, AA, AA = 6 Denominator = Divide out order!

What is the theoretical probability of throwing heads? Jim tossed a coin 35 times. He recorded whether the coin landed on heads or tails. Use the results in the table to find each experimental probability. HEADS TAILS llll llll llll llll P (heads) =________ 2. P (tails) = ________ What is the theoretical probability of throwing heads? 4. tails?

5. What is the difference between the experimental probability of throwing heads and the theoretical probability of throwing heads? (write answer as a percent rounded to the nearest whole percent) This is about 7% Experimental - theoretical

6. What is the difference between the experimental probability of throwing tails and the theoretical probability of throwing tails? (write answer as a percent rounded to the nearest whole percent) This is about 7% Theoretical - Experimental

Mrs. Smith has a bag that contains 6 red Lifesavers, 5 yellow Lifesavers, 8 orange Lifesavers and 9 white Lifesavers. She has another bag that contains 3 Butterfingers, 6 Snickers, and 4 Hershey Bars. You are a student that is being rewarded for good behavior so you get to draw a piece of candy from each bag. Find the following probabilities. Candy Bars 3-Butterfingers 6 Snickers 4 Hershey Bars Lifesavers 6-red 5 yellow 8 orange 9 white

P(yellow Lifesaver, a Snickers) Lifesavers 6-red 5 yellow 8 orange 9 white Candy Bars 3-Butterfingers 6 Snickers 4 Hershey Bars P(yellow Lifesaver, a Snickers) P(red or orange Lifesaver, a Hershey Bar) 9. P(no white lifesaver, a Butterfinger)

David rolled a number cube, with the numbers 1-6 on it David rolled a number cube, with the numbers 1-6 on it. He recorded the results in the table below. Number Outcomes 1 IIII 2 III 3 4 IIII I 5 II 6 I Find the THEORETICAL probabilites: P(4) = ______________ 11) P(odd number) = __________ 12) P(2 or 5) = __________ 13) P(a factor of 3) = _________ 1/6 ½ 1/3 1/3 14) What is the difference between the experimental probability of rolling a 2 or 5, and the theoretical probability of rolling a 2 or 5? 1/3 – 1/4 = 1/12