Simulations and Random Numbers
Take a quick look at the numbers below and, without thinking, pick a number:
Did you pick 2 or 3? If so, you're not alone! A well-known study has shown that if you are quickly shown the numbers 1 to 4 and asked to pick one at random, 75% of people will choose 3, 20% will pick 2 or 4, and only 5% will choose 1.
It seems that humans are not well designed to choose "random" numbers! Instead, we often use computers to generate random numbers for us using algorithms. These kinds of random numbers are called psuedo-random because they are predictable (by nature of having a computer algorithm that creates them). True randomness is not predictable. In fact, some truly random numbers are generated using natural phenomena such as atmospheric noise!
Even though psuedo-random numbers are not truly random, these numbers are often "random enough" to be used in simulations of probability events.
A simulation is a way of reproducing a situation using a computer to simulate the outcomes of a mathematical model for that situation. Simulations are very useful in probability because they allow for an event to be run many times very quickly, and can be used to find experimental probabilities which more closely approach the theoretical probabilities.
Heads and Tails
- Flip a coin 10 times and record the results. Calculate the experimental probability of head or tails. What do you notice?
- Repeat by flipping 30 times. What did you expect to happen to the experimental frequency?
- Use a computer program, like Excel or your CAS calculator, to simulate flipping a coin 100 times. What happens to the experimental frequency? Run this simulation a number of times. Does the more likely outcome change each time? Is this expected?
- Now simulate 1000 flips. What happens to the experimental frequency? What would happen if you simulate 10,000 flips?
Match!
- Shuffle two 52-card decks and draw one card from each, placing them face up. They are a match if they are the same number or face card (Jack, Queen or King). Record the number of matches until all the cards in each deck have been turned over. Calculate the experimental probability of zero matches, one match, two matches etc. Do this 10 times, shuffling the decks between each.
- What was the most likely number of matches you found? Do you notice any patterns in the distribution?
- Simulate this experiment 100 times using random numbers (you will have to decide how you allocate the random numbers for each outcome). What do you notice about the experimental probability of different numbers of matches?
- Simulate the experiment 100 times. Did the experimental probability change? Is this what you expected?
- Predict the theoretical probabilities for each number of matches between two decks of 52 cards.
- Optional: Use your knowledge of probability to determine the theoretical probabilities and compare to your predicted probabilities.