Section 5.3 – basics of simulation

• Simulation – the imitation of chance behavior, based on a model that accurately reflects the experiment under consideration– an effective tool for finding likelihoods of complex

results once we have a trustworthy model.– Gives us good estimates of probabilities

Steps of Simulation

1) State problem or describe experiment2) State the assumption3) Assign digits to represent outcomes4) Simulate many repetitions

• State the problem or describe the experiment.– Toss a coin 10 times.

– What is the likelihood of a run of at least 3 consecutive heads or 3 consecutive tails?

• State the assumptions– A head or tail is equally likely to occur on

each toss

– Tosses are independent of each other• What happens on one toss does not influence

the next toss

• Assign digits to represent outcomes.– One digit simulates one toss of a coin– Odd digits represent heads; even represent tails.

• Simulate many repetitions.– Look at 10 consecutive digits in Table B simulates

one repetition– 19223 95034 05756 28713 96409 12531– hhh hhh t tt hhh ttt hhh– Repeated 25 total times, 23 had a run of 3 or more

• State your conclusions. – Estimate probability of run by proportion– 23/25 = 0.92

Assigning Digits• Choose a person at random from a group of which

70% are employed. One digit simulates one person.– 0, 1, 2, 3, 4, 5, 6 = employed– 7, 8, 9 = not employed– 1, 2, 3, 4, 5, 6, 7 = employed– 8, 9, 0 = not employed– 00, 01, 02,…69 = employed– 70, 71, 72,…99 = not employed– 01, 02, 03,…70 = employed– 71, 72, 73,…99, 00 = not employed– Good Options? Bad Options?

Assigning Digits more practice…• Choose on person at random from a group of

which 73% are employed. – Now 2 digits are needed to simulate one person– 00, 01, 02, … , 72 = employed– 73, 74, 75, … , 99 = not employed– 01, 02, 03, … , 73 = employed– 74, 75, 76, … , 99, 00 = not employed

Assigning Digits even more practice…• Choose one person at random from a group

50% employed, 20% unemployed, and 30% are not in the labor force.– 0, 1, 2, 3, 4 = employed– 5, 6 = unemployed– 7, 8, 9 = not in work force– 1, 2 = unemployed– 3, 4, 5 = not in labor force– 6, 7, 8, 9, 0 = employed– Lots of options here!

Frozen yogurt sales

• State the problem.– Simulate 10 fro yo sales based on the recent history of

38% Chocolate, 42% Vanilla, 20% Strawberry• State the assumptions– The pairs of digits on the random digit Table B are

independent of each other• Assign digits– 01, 02, … 38 = Chocolate (C)– 39, 40, … 80 = Vanilla (V)– 81, 82, … 99, 00 = Strawberry (S)– There are other options here!

• Simulate– Start at line 133– 45740 41807 65561 33302– 45 74 04 18 07 65 56 13 33 02– V V C C C V V C C C

• Conclusions

Randomizing with the Calculator

• randInt – TI 83 – MATH/PRB/5:randInt– TI 89 – Catalog / F3 (flash apps)– randInt (1, 6, 8)• Rolling a die 8 times

– randInt (0, 99, 12)• Choosing 12 two digit numbers between 00 and 99

– randInt (1, 2, 10)• Flipping a coin 10 ten times

examples

• #59. state how you would use the following aids to establish a correspondence in a simulation that involves a 75% chance– A coin– A six-sided die– A random digit table– A standard deck of playing cards.

• Homework 61, 62, 70, 71, 56