cs1512 jhunter/teaching/cs1512/lectures/ 1 cs1512 foundations of computing science 2 lecture 22...

www.csd.abdn.ac.uk/~jhunter/teaching/CS1512/lectures/

CS1512

CS1512

1

CS1512Foundations of

Computing Science 2

Lecture 22

Digital SimulationsProbability and statistics (3)

© J R W Hunter, 2006


CS1512

CS1512

2

Simulations

Programs regularly used to simulate real-world activities.• city traffic• the weather• nuclear processes• stock market fluctuations• environmental changes

• shipping movements in the Straits of Dover!


CS1512

CS1512

3

Simulations

Simulations are normally only partial and involve simplification.

Greater detail has the potential to provide greater accuracy.

Greater detail typically requires more resource:• Processing power.• Simulation time.


CS1512

CS1512

4

Benefits of simulations

Support useful prediction.• e.g. the weather

Allow experimentation.• Safer, cheaper, quicker.

Example:• ‘How will the wildlife be affected if we

cut a highway through the middle of this national park?’


CS1512

CS1512

5

Basic Algorithm

set up a number of objects to represent the world that you are trying to model

add the objects to a collection maintained by the simulation

set the simulation time to zero

while the simulation time isn’t up

for each object

• call the method that executes the behaviour of the object during the simulation time step

• do any necessary output

increment the simulation time by the time increment

wait for an amount of real time corresponding to the time increment


CS1512

CS1512

6

ChanSim: Main simulation loop

public void simulate(int timeStep, int speedUp) { while (time < totalSimulationTime) { for (int i = vessels.size()-1; i >= 0; i--){ Vessel vessel = (Vessel) vessels.get(i); if (vessel.onChart()) vessel.erase(chart); vessel.move(timeStep); if (vessel.onChart()) vessel.draw(chart); } chart.update(); time = time + timeStep; wait(timeStep*60*1000/speedUp); //*60*1000 to convert minutes to msec }}


CS1512

CS1512

7

Vessel: move

/**

* Determine where the vessel will be after timeStep

*/

public void move (int timeStep)

{

// distance travelled in km per time step (in minutes)

double delta = speed*timeStep/60;

location = new Location(location.getx()+

Math.sin(bearing)*delta,

location.gety()-

Math.cos(bearing)*delta);

}


CS1512

CS1512

8

Probability


CS1512

CS1512

9

Initial thoughts

A person tosses a coin five times. Each time it comes down heads. What is the probability that it will come down heads on the sixth toss?

½ or ‘1 chance in 2’ or 50%• assumed that the coin is ‘fair’• ignored empirical evidence of the first five tosses

less than ½ or ...• assumed that the coin is ‘fair’• thought about ‘law of averages’ – in the long run, half heads, half tails• possibly confused about the question - which was not “What is the probability that there will be six heads in a row?”

more than ½ or ...• cynical – assumed two headed (or biased) coin


CS1512

CS1512

10

Definition

Probability: the extent to which an event is likely to occur, measured by the ratio of the number of favourable outcomes to the total number of possible outcomes.

10 balls in a bag 7 white; 3 red

• close eyes• shake bag• put in hand and pick a ball• 10 possible balls (outcomes)

Can you think of any reason why any one ball should be picked rather than any other?

If not, then all outcomes are equally likely.


CS1512

CS1512

11

Definitions

Probability of picking a red ball

• favourable outcome = red ball

• number of favourable outcomes = 3

• number of outcomes = 10

• probability = 3/10 (i.e. 0.3)

Note:

• probability of picking a white ball is 7/10 (0.7)

• probabilities lie between 0.0 and 1.0

• we have two mutually exclusive outcomes (can’t be both red and white)

• outcomes are exhaustive (no other colour there)

• in this case the probabilities add to 1.0 (0.3 + 0.7)

• a probability is a prediction


CS1512

CS1512

12

More definitions

Trial

• action which results in one of several possible outcomes;

• e.g. picking a ball from the bag

Experiment

• a series of trials (or perhaps just one)

• e.g. picking a ball from the bag twenty times

Event

• a set of outcomes with something in common

• e.g. a red ball


CS1512

CS1512

13

Probability derived a priori

Suppose each trial in an experiment can result in one (and only

one) of n equally likely (as judged by thinking about it)

outcomes, r of which correspond to an event E.

The probability of event E is:

rP(E) = ― n

a priori means “without investigation or sensory experience”


CS1512

CS1512

14

More complex a priori probabilities

Probability of throwing 8 with two dice:

• 5 outcomes correspond to event “throwing 8 with two dice”• 36 possible outcomes• probability = 5/36

1 2 3 4 5 6 1 2 3 4 5 6 7

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10 11

6 7 8 9 10 11 12


CS1512

CS1512

15

Probability derived from experiment

Toss a drawing pin in the air - two possible outcomes:• point up• point down

Can we say a priori what the relative likelihoods are? (c.f. ‘fair’ coin)

If not, then experiment.

Probability based on relative frequencies (from experimental data)

If, in a large number of trials (n), r of these result in an event E, the probability of event E is:

rP(E) = ― n


CS1512

CS1512

16

Probability as a relative frequency

• number of trials must be ‘large’ (how large?)

• trials must be ‘independent’ - the outcome of any one toss must not depend on any previous toss

• no guarantee that the value of r/n will settle down

• compare with the relative frequencies for existing data• if we have enough experiments then we believe that the

relative frequency contains a general ‘truth’ about the system we are observing.

cs1512 jhunter/teaching/cs1512/lectures/ 1 cs1512 foundations of computing science 2 lecture 22...

Documents

probability slide

time increment slide

possible outcomes probability

red ball slide

number of outcomes

prediction slide

white outcomes

probabilities probability