Section 5 (4 Credits)

Random and Probability – It is all about chance.

A random event has an unpredictable outcome. There is a lot of probabilistic language associated with probability and the value we assign to these words varies widely.

(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.

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”.

Language

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 of them all.

 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

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%.

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

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

ANALYSIS

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 54% of “Pin Up” outcome. The graph starts to show stability and another 50 throws would give a more reliable answer.

C = This investigation suggests the probability of tossing a drawing pin and having it land point up is about 54%. The graph shows some re\liability in this estimate as being 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.

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

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).]

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.

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

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.

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 1,2,3,4,5,6,7,8,9 and 10. These are set up as skittles are in a triangle and crossed out when answers are created. EG 2,3,5 on a dice gives 2x3–5 = 1.

7                8                9                10

4                5                6

2                3

1

Summary

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.

Probability language is a complex mix of creed and experience.

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 any one buys a ticket let alone the millions sold each week.

Horse racing is plagued with cheats and casinos play knowing for all the money that is gambled about 4% must go to them. People continue to play the odds.

Assessment

Section 5 (4 Credits)

Random and Probability – It is all about chance.

xxxx

Find

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.