Section 5 (4 Credits)

Random and Probability – It is all a bit of a chance.

two dice
 

A random event is some occurrence or happening with an unpredictable outcome. The most common example is probably the toss of a coin or the throw of a couple of dice. There are countless chance events in our human world. We experience real-time probability for example whenever a piece of buttered toast gets flipped onto the floor or when we buy a LOTTO ticket. One of those events is almost predictable.


The language associated with probability is almost intuitive and the value we assign to the words and phrases varies widely. See below. Chance language ideas develop from early days.

Firstly...

Task 1

(a)   What is the number that represents an event that MUST happen or be true. Eg You are alive yesterday.

(b)  What is the number that represents an event that can never happen. Eg You were dead yesterday.

The actual numbers here are zero for “can never happen” and one for “must happen”. All other probabilities fall between 0 and 1.


[This is another example of "1 can be anything I choose one to be". We could have chosen 360 for "It will happen" . I think we were over the use of 360 by then however. Blaise Pascal and Pierre de Fermat were the two key developers of probability ideas and this was due to the need to understand gaming and probably decided upon 1.

A coin toss for example has two equally likely outcomes. Those are H or T. The probability of a head, Pr(H) = ½ = 0.5 = 50% or “a half” or “fifty-fifty”.


Tossing the die is an "experiment" and the experiment has an "outcome". This outcome is one of all possible outcomes or "sample space". Each outcome has a probability. The sample space for a fair six sided die is {1,2,3,4,5,6} and each has a probability of 1/6th of happening. The probability of the "event" of "tossing an even number" on a die is 3/6 or 1/2.


Language

Task 2

List 20 words in common use that have a probability associated with them. Eg Evens.

 

Here is a table to add your words and estimate the probability associated with all of them all. Compare and discuss with another person.

 

Evens

0.5

Might

 

Must

 

Likely

 

Never

 

Pretty likely

 

Probable

 

Could

 

Uncertain

 

Possible

 

Will

 

Almost never

 

Certain

 

Almost Certain

 

Half

 

Unlikely

 

Beyond reasonable chance

 

On the balance of probabilities

 

 Maybe

 

 

 

 probably

 

 

 

 

 

 

 

 

        The curious thing about the estimates is another person will give you a different response. Is it a surprise to you that a District Court Judge estimated “beyond reasonable doubt” as 51%. My guess would have been 90% or better!

 
          We use fractions between 0 and 1 to represent probabilities. These fractions can be in decimal form so 1/2 = 0.5 = 50% or the notion of evens.

The language of probability includes words such as “fair”, “outcome”, “experiment”, “event”, “random generator”, “tool” and “expectation”. A spreadsheet is a wonderful place to explore probability using the function “RAND(0)”. This function generates a random number between 0 and 1.

 The spreadsheet is actually a wonderful place to explore mathematics! It is quite surprising how many problems can be solved using probability on a spreadsheet as well. Simulating outcomes from large runs on a spreadsheet is used to solve some very difficult problems.

For example: If I toss three dice what is the probability I can form a triangle using these numbers as measures of the sides. EG (1,1,6) and (2,2,4) do not but (3,4,5) and (2,2,2) do.

 

A Statistical Investigation

All probability experiments should be shaped as a statistical investigation using the pseudonym PPDAC.


            P = Problem or Question

            P = Plan or method

            D = Data or the results of your plan

A = Analysis or unpacking and making sense of the data

C = Conclusion or Answer to the Question asked with elaboration.

 

A Simple Example of a Statistical Investigation in Probability

P = I wonder what the probability of throwing a drawing pin and having it land point up is.

P = I will toss the drawing pin onto a table from about 30cm and repeat this experiment 50 times. I am throwing it 50 times because I think that will give me a stable answer. The drawing pin could land on its back or side both of which are not point up.

D =      D,D,D,D,U,D,D,U,D,U,D,U,D,D,U,U,U,U,D,D,U,U,D,D,D

            D,U,U,D,U,U,U,U,D,U,U,U,U,U,U,D,U,U,U,D,D,D,U,U,D

 

A

The number of D outcomes was 23 and the number of U outcomes was 27. The longest run of D’s was 4 and for U’s was 6. The total number of throws was 50. This makes the probability of throwing a drawing pin point up from this experiment 27/50 or 54%.

 

A graph of this data (using Excel) shows the trending 50% of “Pin Up” outcome. The graph starts to show stability and another 50 throws would give a more reliable answer.

Simulation
          Graph

 

C = This investigation suggests the probability of tossing a drawing pin and having it land point up is about 50%. The graph shows some reliability in this estimate and suggests slightly above 50%. The margin of error is 1/√50 = 14% suggesting a lot of variation is in the answer and a better estimate would be 54%±14% so the actual answer is very likely to be between 40% and 68%. This interval of error could be reduced by throwing the pin many more times. A good estimate would be about half. 

 

Task 3

Create an experiment like this one and investigate the probability of an outcome.

 

Suggestions: “getting a 1 on a die, tossing a head on a coin, toast landing buttered side down”. Have some fun!

 

Returning to a task in Section 2

Task 4
Imagine you are a coin and you flip yourself 100 times. Record ten rows of ten outcomes you create. One the other grid toss a coin 100 times and record the outcomes. Give the coin a good toss [or use a die and record odds (H) and evens (T).]

 

Grid   Grid

 

Look carefully at the two tables and see if you can spot a key event that happens events.

This task hopefully illustrated how un-random we are as humans. It would be a very unusual person who without prior knowledge guessed a long run of 5 or 6 Heads in a row. The overwhelming urge in all of us is to preserve a balance and knowing that the outcome is 50/50 we might get 2 or 3 in a row but very seldom more than this. Being random however allows for much longer runs. No notion of the previous outcome is taken into account for the next.

 

Games

A die or two dice or more bring the element of chance to the table and create an astonishing number of very cool games and fun.  One die

 Monopoly
              image

Monopoly is the game to introduce buying and selling of property, tax, penalties and rewards. This is an essential game and learning for all young minds and is fun"

 

 Yahtzee
            Scoring sheet

 


Yahzee is a number game using 5 dice. It involves getting patterns such as runs of 5’s, a full house, two pairs and so on. Points are awarded for scoring and a total determines the winner. This score sheet can be enlarged and printed. A Yahzee is 5 numbers of the same type. Each player has three tosses and selects the dice to retain before the second and third throws. There is a variety of this game I invented on my website called Numatzee using 6 ten sided dice and teh target is patterns in numbers such as "the powers of 2". It is a lot of fun and learning.

 

Dungeons and Dragons is a much more complex game using many different dice and a complex system of gaining and loosing strength. The goal being to overcome the opponent!

 

A simple game using three dice to help develop number reasoning is called Skittles. The three dice are tossed and the resulting numbers used to get the answers 1 to 12.  EG 2,3,5 on a dice gives 2x3–5 = 1. I make students write the reasoning on the whiteboard by their answer. And, I give extra points for using x, (), divide and powers.


skittles
            game
 

Probability extends into genetics, nuclear physics, chaos theory and finance. Gaining a working knowledge of a how a spreadsheet works will open up a world of solutions and fun.

 

 Lotto is played by most Kiwis every week and they all know the chance of winning is less than 1 in 4 million. One in 100 is considered impossible. 1 in 4 million is so screamingly small it is a wonder anyone buys a ticket let alone the millions sold each week.

 


Statistics Connection
 Statistics and probability are strongly linked. We can select a sample from a population in many ways. One of the most reliable ways of getting a representative sample is to take a random sample and random means probability. The data is ordered and a random generator is used to choose the data to analyze and learn more about the population.


Why take a sample? A population is often huge and can not be collected so a sample is the best we can do. We usually do not know what the population parameters like mean and spread are so we take a sample to find out. The measurements from last years crop is all we have so that is the sample. A sample mimics the population and so is a representative. More on this in the next section but a problem for probability is "how big should a sample be?". This can be answered using probability.


In this image is a graph of sample size versus spread or width  of the data. When selecting just a few data points they can come from left, right or center and they do just that when selected randomly. The result is there is a lot of variation or spread. As the sample size gets bigger data from left and from right and from center is just as likely to turn up and the variation across the data gets to be less. The problem is where to draw the line between expensive sample and time consuming analysis to get the answers that can be relied upon.


Task 4
Look at the graph and decide what sample size you would choose.


sample size


The clue is in the shape and the bend of the curve. The variation does not decrease a lot once 50 to 100 have been selected. Taking 250 does improve reliability but not a lot more than 50. Hence my eye "says" a reliable sample would selected if it was between about 30 to 60 in size. The curve also shows evidence for the rule that "four times the sample size will half the variation". This is not obvious at all but is correct and validated by experiment.


Expected Value
Just how much can you expect as a return by buying a raffle, lotto or scratchy? This was the problem Fermat and Pascal battled through. The answer is quite simple now.


Here is a simple game. Place a bet of $2 on the outcome of throwing three coins. If tall three coins are the same, HHH or TTT then I will give you $4 as a return. Is the game fair and how much can I expect to win?


The possibilities are {HHH, HHT, HTH, THH, HTT, TTH, THT, and TTT} or 8 possibilities. The Expected Return is 1/8th x $4 + 1/8th x $4 for the HHH and TTT outcomes and $0 for all others. This means I can expect a $1 return for every $2 bet. Clearly this is not a fair game. It would be fair if I could expect $2 back for every $2 bet.


Here is an experiment showing 20 randomly generated triple tosses.
                                                                                                                    random
                tosses

Looking down the list of tosses there are 6 payouts of $4 or $24 for a total collection of $40. The game is not fair! The odds are heavily in favor of the person taking the bets. In this short 20 games the person makes $16. Not bad for about 10 minutes work.



Probability can be used to solve this puzzle as well.
Triangle
            problem

The puzzle is what percentage of the square is shaded?


If we imagine throwing random darts at the square some will be inside teh shaded area and the others will not be. The proportion of darts inside to all darts thrown is the answer to the problem. A little bit of spreadsheet knowledge will find a pretty good estimate of the answer.


You could draw the shape as a 1m square on the floor on the classroom and put rulers as borders. Then randomly scatter a bucket of 100 marbles, or peas, or rice or dice into the shape. Counting the number that lie inside the shaded shape will give the percentage shaded.

Then as I pondered I thought the answer can be found by cutting out the shapes and weighing the pieces. Suitable scales can be purchased from the Warehouse or borrowed from the Science Dept. Resene Paints gave all schools a 1m square of fairly heavy lino material.

Assessment

Section 5 (4 Credits)

Random and Probability – It is all about chance.


1. Do Task 5. What is your estimate of the percentage of the square that is shaded?

2. Breakout! What is a good strategy to use to win this game?

BREAKOUT… a Game for Two Players

EQUIPMENT: Prison Block layout  below, ten counters or soft toy prisoners, 2x 1 to 6 dice.

 

COVID19 PRISON BLOCK

Cell Block A

Cell Block B

Cell 0

 

 

Cell 0

 

 

Cell 1

 

 

 

Cell 1

 

 

 

Cell 2

 

 

 

Cell 2

 

 

 

Cell 3

 

 

 

Cell 3

 

 

 

Cell 4

 

 

 

Cell 4

 

 

 

Cell 5

 

 

 

Cell 5

 

 

 

Cell 6

 

Cell 6

 

 

 

You can draw this on a big sheet of paper or set up the living room or even print this sheet.

 

INSTRUCTIONS : Place your 10 prisoners (counters/soft toys) in the cells. You can spread them put or put them all in one or any other combination. One payer puts their prisoners in Cell Block A and the other in Cell Block B.

 

THE FUN PART

Now decide who starts somehow and taking turns toss both dice. Subtract the two numbers to find the difference (big – small) and if you have a prisoner(s) in that cell then let one of them out. For example if I throw a 3 and a 5, the difference is 5-3=2 so I release a prisoner form Cell 2 in my Block. Repeat until one player has released all his prisoners and wins the game.

 

HINTS: Play the game a couple of times to get the idea. Be a mathematician and look for patterns.

Well done, five sections over! On to Section 6! Is the course fun? Use the Navigator to find the next section.