1.3 Simulations and Experimental Probability (Textbook Section 4.1)

 One of the major branches of modern mathematics in the “mathematics of chance”, commonly called probability.  Probability is a number that gives a measure of how likely an event is to occur  Probability Theory, deals with chance, random variables, and the likelihood of outcomes.  Random Variables - can have any of a set of different values. In statistical applications it is denoted by a capital letter, most commonly X, while its individual values are denoted by the corresponding lower case letter.

 Experiment – a sequence of trials in which a physical occurrence is observed  Trial - one repetition of an experiment  Outcome – the result of an experiment  Sample Space – the set of all possible outcomes (sometimes referred to as the universal set )  Event – a subset of the sample space – a possible outcome of an experiment

 Fair Game – a game in which all players have an equal chance of winning (and losing!)  Simulation – an experiment that models an actual event  In situations with many variables, it can be difficult to calculate exact values; in such cases simulations can provide a good estimate.  Simulations can also help verify theoretical calculations.  The Law of Large Numbers – The more times an experiment is repeated, the closer the outcome should be to the theoretical probability

 Experimental Probability – the observed probability of an event  Theoretical Probability – the probability of an event deduced from analysis of the possible outcomes  An experiment can be simulated using random numbers. The random numbers can be generated using a calculator, tables, or computer spreadsheet.

Experiment #1  Suppose each morning you and a friend play a simple game to see who pays for coffee. If the outcome is “heads”, your friend pays for both coffees. If the outcome is “tails”, you pay for both. You do this 5 times a week.  In groups of two or three,  Predict the amount you would pay after 4 weeks.  Toss a coin and record the outcomes and the amount paid after 1 toss. Repeat 20 times.  Determine the experimental probability and the amount paid after 20 trials (Assume $1 per coffee)

Experiment #2  Roll a die and record the outcome. Repeat the process 50 times. Find the experimental probability of rolling a “blue” after 10, 20, 30, 40 and 50 times. # of Trials# of Blues

Experiment #3  Using a graphing calculator, repeat Experiment #2.  Assign  Blue = 1  Red = 2  Yellow = 3  Green = 4  MATH  PRB  5:randInt(1,4,5) # of Trials# of Blues

Experiment #4  Suppose Nicole has a batting average of This indicates 32 hits in every 100 attempts (or 8 hits in 25 attempts). Use a simulation to estimate the likelihood that she has no hits in a game – assume there are 3 at-bats per game. Solution  Fill a container with 25 slips of paper, 8 which have the word HIT written on it. Draw one slip and record. Make sure to return the slip before you draw again (and shake up the container!). Repeat the draw a total of 3 times to simulate one game.  After recording 30 game simulations, count the number of outcomes that satisfy the condition and calculate the experimental probability.