Investigation: Using statistics and probability in the real world

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

1. Flip a coin 10 times and record the results. Calculate the experimental probability of head or tails. What do you notice?
2. Repeat by flipping 30 times. What did you expect to happen to the experimental frequency?
3. 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?
4. Now simulate 1000 flips. What happens to the experimental frequency? What would happen if you simulate 10000 flips?

Match!

 

1. 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. 
2. What was the most likely number of matches you found? Do you notice any patterns in the distribution?
3. 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?
4. Simulate the experiment 100 times. Did the experimental probability change? Is this what you expected?
5. Predict the theoretical probabilities for each number of matches between two decks of 52 cards.
6. Optional: Use your knowledge of probability to determine the theoretical probabilities and compare to your predicted probabilities.

 

Prize time

A company sells bottles of water and each bottle cap has one of 4 possible letters written on the inside. The company offers a prize if you collect all 4 of the letters. We will assume that each of the 4 letters is as likely as any other letter. In other words, there is a 1 in 4 chance of getting any letter.

1. What do you estimate to be the minimum number of bottles you would need to buy to have all 4 letters (if you are not allowed to swap letters with others)?
2. Design a way to simulate the experiment using technology or dice.
3. Carry out the simulation 40 times to work out how many bottles need to be bought to collect all 4 letters.
4. Record the results in the frequency table below.

Number of bottles	4	5	6	7	8	9	10	11	12	13	14	\gt 14	Total
Frequency													40

5. What was the most common number of purchases required to collect the set? How does this compare to your earlier estimate?
6. Using the table you have generated above, what is the probability of a person completing the word in less than 10 purchases of bottled water?

 

Baby boom

A recently married couple wish to start a family. They have decided they definitely want at least one boy and one girl. They have agreed to continue having children until they have at least one of each sex.
e.g. the family may have boy boy boy girl
which means they had 4 children before they achieved a boy and girl.

1. Design a simulation to determine the average number of children they will have to have before achieving a boy and a girl. Run the simulation 50 times.
2. Create a graph of your results.
3. What is the average sized family required? What was the most common sized family required?
4. What is the probability that the family will have 3 or less children?
5. What real-life complications may lead to this simulation to not represent the true probabilities for the couple?
6. Scenario: "A country with a one-child policy and a strong societal pressure for having male children, relaxes the policy to allow families to try until they have a male child." Would we see a doubling in the population? Can you alter the experiment above to explore this case?

 

Penalty points

From his previous games, it is known that Kevin scores 80% of the penalties shots he takes.

1. Estimate how many penalties will it take Kevin to successfully take 6 shots? (not necessarily in a row).
2. Using random numbers generated between 1 and 10, describe how you could simulate an 80% chance of something happening. 
3. Design a simulation and run it 40 times and record your results in a frequency table.
4. From your simulation, determine the average number of penalties taken before Kevin scores 6 penalty shots,
5. What is the probability that Kevin takes 10 or less shots to achieve shooting 6 successfully?

6. What real-life complications may lead to the simulation performed not representing the true probabilities for the situation?