Ex #1: Using random numbers to simulate the gender of ...



Chapter 5.3

A simulation technique uses a probability experiment and random numbers to mimic a “real-life” situation. It does not give exact results. The more times the experiment is performed the closer to the actual results should be to the theoretical results. (This is the theory of large numbers).

Studying the actual situation may be too costly, too dangerous and too time consuming.

Ex: Flight simulators to practice engine fires, engine outs, depressurization and etc.

Steps to experiment simulation:

1. List all possible outcomes of the experiment

2. Determine the probability of each outcome

3. Set up a correspondence between the outcomes of the experiment and the random numbers

4. Select random numbers and conduct the experiment

5. Repeat the experiments and tally the outcomes.

6. Compute the statistics

7. State the conclusion or recommendation.

Step 3 explanation: Tossing a coin can be simulated by using random numbers. Since there are only two outcomes and each has the probability of 1/2, odd digits can be used to represents heads and vice versa.

A rolling of a die can also be simulated using random numbers 1-6 ignoring 7-9 and 0.

Mathematical simulation techniques use probability and random numbers to create conditions similar to those of real-life situations. Computers play a major role in simulation.

Ex #1: Using random numbers to simulate the gender of children born.

Solution: There are only 2 possibilities, male and female. The probability of each outcome is ½ the odd digits can be used to represent male births and the evens simulate female births.

Ex #2: Using random numbers simulate the outcomes of a tennis game between Bill and Mike, with the additional condition that Bill is twice as good as Mike.

Solution: Since Bill is twice as good as Mike, he will win approx. two games to one. Hence, probability Bill wins is 2/3 and the probability Mike wins is 1/3. Random digits 1-6 can be used to represent a game bill wins; and random digits 7-9 can be used to represent Mike’s wins. Suppose they play five games, and the random number selected is 86314. This means that Bill won games 2, 3, 4, and 5. Mike won the first game.

Ex #3: A die is rolled until a 6 appears. Using simulation find the average number of rolls needed. Try the experiment 20 times.

Solution:

Step 1: List all possible outcomes, 1, 2, 3, 4, 5, and 6.

Step 2: Assign the probabilities. Each outcome has a probability of 1/6.

Step 3: Set up correspondence between the random numbers and the outcome. Use random numbers 1-6 and omit 7-9, and 0.

Step 4: Select a block of random numbers and count each digit 1 through 6 until the first 6 is obtained. For example block 857236 means that it takes 4 rolls to get a six.

Step 5: Repeat the experiment for 19 numbers and tally the data.

Step 6: Compute the results and draw a conclusion. In this case you must find an average which is the number of rolls divided by 20 which is the number of trails.

Ex #4: A person selects a key at random from four keys to open a lock. Only one key fits. If the first key does not fit, she tries other keys until one fits. Find the average number of keys a person will have to try to open the lock. Try the experiment 25 times.

Solution: Assume that each key is numbered 1-4 and that key 2 fits the lock. For simulation select a sequence of digits, using only 1 through 4, until digit 2 is reached.

The theoretical average is 2.2.

Ex#5: A box contains five $1 bills, three $5 bills and two $10 bills. A person selects a bill at random. What is the expected value of the bill? Perform the simulation 25 times.

Solution: List all possible outcomes. They are $1, $5 and $10. Assign the probabilities to each outcome. P($1)=5/10 P($5)=3/10 p($10)=2/10

Set up a correspondence between the random numbers and the outcomes. Use ransom numbers 1-5 to represent a $1 bill being selected, 6-8 to represent a $5 bill being selected, and a 0 or 9 to represent a $10 bill being selected

Select 25 random numbers and tally the results

Number Results($)

4 5 8 2 9 1 1 5 1 10

2 5 6 4 6 1 1 5 1 5

9 1 8 0 3 10 1 5 10 1

8 4 0 6 0 5 1 10 5 10

9 6 9 4 3 10 5 10 1 1

Compute the average: [pic]

Hence, the expected value is $4.64

If we use the expected value formula:

[pic]

Remember the simulation technique does not give accurate results. The more times the experiment is performed the closer to the theoretical results it will be (law of large numbers).

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