# True Probability vs experimental probability vs theoretical probability

True Probability vs. experimental probability vs. theoretical probability

Watch the Numberphile Video “How random is a coin toss” https: //youtu. be/AYn. Jv 68 T 3 MM

TRUE probability

What is the Probability of flipping a coin and getting Heads Theoretical estimate

What is the Probability of flipping a coin and getting Heads Theoretical estimate Never flipped a coin, never did an experiment, its all a construct in my brain What assumptions did I make? • Only 2 outcomes • Equally likely

What is the Probability of flipping a coin and getting Heads Theoretical estimate Never flipped a coin, never did an experiment, its all a construct in my brain What assumptions did I make? • Only 2 outcomes • Equally likely

What is the Probability of flipping a coin and getting Heads Theoretical estimate Experimental Probability Never flipped a coin, never did an experiment, its all a construct in my brain Flipped a coin 100 times, 49 came up with Heads. What assumptions did I make? • Only 2 outcomes • Equally likely I did an experiment/data was collected in order for me to produce the probability

What is the Probability of flipping a coin and getting Heads Things we need to know How well do we know them How many revolutions per second is the coin turning at What velocity did the coin get flipped at Is the length of the cylinder uniform Is the cross section perfectly circular Is the coin equally weighted Does it produce only 2 outcomes Yes

Theoretrical Estimate for Getting Heads: P(H)=0. 5 Things we need to know How well do we know them How many revolutions per second is the coin turning at What velocity did the coin get flipped at Is the length of the cylinder uniform Is the cross section perfectly circular Is the coin equally weighted Does it produce only 2 outcomes Minting machine should be pretty high quality, so pretty much (but not eactly) Yes

What is the Probability of flipping a coin and getting Heads Things we need to know How well do we know them How many revolutions per second is the coin turning at What velocity did the coin get flipped at No Is the length of the cylinder uniform Minting machine should be pretty high quality, so pretty much (but not eactly) Is the cross section perfectly circular Is the coin equally weighted Does it produce only 2 outcomes Yes

What is the Probability of flipping a coin and getting Heads Things we need to know How well do we know them How many revolutions per second is the coin turning at No What velocity did the coin get flipped at No Is the length of the cylinder uniform Minting machine should be pretty high quality, so pretty much (but not eactly) Is the cross section perfectly circular Is the coin equally weighted Does it produce only 2 outcomes Yes

What is the Probability of flipping a coin and getting Heads Things we need to know How well do we know them How many revolutions per second is the coin turning at No What velocity did the coin get flipped at No Minting machine should no one can control the velocity or be pretty high Is the cross section perfectly circular revolutions per second, so that isquality, pureso pretty much Is the coin equally weighted chance (but not eactly) Is the length of the cylinder uniform Does it produce only 2 outcomes Yes This estimate will be very close to the true probability

Things we need to know How well do we know them How many revolutions per second is the coin turning at More and less common possibilities considered What velocity did the coin get flipped at More and less common hand speeds considered Is the length of the cylinder uniform Is the cross section perfectly circular Is the coin equally weighted Does it produce only 2 outcomes Minting machine should be pretty high quality, so pretty much (but not eactly) Yes

Things we need to know How well do we know them How many revolutions per second is the coin turning at More and less common possibilities considered What velocity did the coin get flipped at More and less common hand speeds considered Is the length of the cylinder uniform This estimate will be even closer Is the cross section perfectly circular probability Is the coin equally weighted Does it produce only 2 outcomes Minting machine to thebe true should pretty high quality, so pretty much (but not eactly) Yes

Things we need to know How well do we know them How many revolutions per second is the coin turning at Measured to 1 dp What velocity did the coin get flipped at Measured to 1 dp Is the length of the cylinder uniform Minting machine should be pretty high quality, so pretty much (but not eactly) Is the cross section perfectly circular Is the coin equally weighted Does it produce only 2 outcomes Yes

This estimate is frighteningly close to the true probability, and you may even choose to think of it as the true probability itself Things we need to know How well do we know them How many revolutions per second is the coin turning at Measured to 1 dp What velocity did the coin get flipped at Measured to 1 dp Is the length of the cylinder uniform Minting machine should be pretty high quality, so pretty much (but not eactly) Is the cross section perfectly circular Is the coin equally weighted Does it produce only 2 outcomes Yes

Things we need to know How well do we know them How many revolutions per second is the coin turning at Measured exactly What velocity did the coin get flipped at Measured exactly Is the length of the cylinder uniform Is the cross section perfectly circular For sure Is the coin equally weighted Does it produce only 2 outcomes Yes

Things we need to know How well do we know them How many revolutions per second is the coin Measured exactly Thisturning is noatlonger a random event What velocity did the coin flipped at (noget uncertainty) Measured exactly Is the length of the cylinder uniform Is the cross sectionit perfectly Therefore is not acircular matter For sure of probability Is the coin equally weighted anymore, the coin will either land as heads Does it produce only 2 or outcomes Yes it wont.

TRUE probability • True probability involves an exact (just about) understanding of all (just about) the factors involved that lead to a certain outcome. • All but doesn’t exist in real life. Rather, it exists as a concept, an ideal that our theoretical and experimental probabilities are working towards.

One of the frequent considerations a student is expected to make is to enumerate reasons why their estimate is not, in fact, the TRUE PROBABILITY

• For Example: • Mr Chida is playing roulette, and he quickly calculates that the odds of winning on the next roll is 24%. • Is this the true probability? Justify your answer. • force of the hand • period of that force • knowledge of inertia • friction of the wheel, weight • smoothness and roundness of the ball • What are the more and less common hand speed during the turning. •

ADVANTAGES and DISADVANTAGES

Advantage of Experimental Probability • An advantage of estimating probabilities using experimental probabilities is that this procedure is relatively free of assumptions. • For example, consider estimating the probability among a population of men that prefer strawberry jam to raspberry jam. • A direct estimate could be found by counting the number of men who satisfy our

Disadvantage • A disadvantage in using experimental probabilities arises in estimating probabilities which are either very close to zero, or very close to one. • In these cases very large sample sizes would be needed in order to estimate such probabilities to a good standard of relative accuracy. • It also takes a lot of time and effort to design your experiment/collect the data.

Advantage of Theoretical Probability • Little cost and effort, but can often involve some mental acrobatics • At the end of our findings, their aren’t a whole lot of things that could have gone wrong Disadvantage of Theoretical Probability • Very easily applied to trivial examples such as card and dice games, but requires serious mental acrobatics to apply it to practical situations or research

Applying our Findings

Application of an Experimental Probability Study the past and apply findings to the future Data was collected from the year Y 9 s at Chida Grammar School in 2015 Y 9 Y 10 Own Device 110 70 180 Doesn’t own device 90 135 225 200 205 405 • • • it was estimated that 55% of next year’s Y 9 s will have their own device What are some of the problems with the conclusion above? It has been 6 years since the data was collected and devices have become cheaper The neghbourhood around the school may have got poorer or richer effecting parents ability to buy devices for their kids Teachers may use devices more since data was collected, meaning students are more likely to spend money getting the device The data collection might be muddled. Some people may say yes to owning a device even though they share it with a sibling This years Y 9 s could be more tech savvy and so they convinced their parents t get them foir a device for no n school stuff

Application of an Experimental Probability Study a cross section and apply it to another cross-section An online survey collected data from the Chidaville Rugby Association • • forwards Backs Access to Gym 1100 1800 2900 No Acess to Gym 500 300 800 1600 2100 3700 it was estimated that the probability a Randomly chosen rugby player in Mount Chida Grammar School has access to a gym is 74% What are some of the problems with the conclusion above? The Chidaville Rugby association includes adult players and they would be more likely to have access to a gym It also depends on how far Mount Chida Grammar School is from the nearest gym, if it is much further than the average distance between a high school and gym these results wouldn’t apply If Mount Chida Grammar School is in a poor area, much poorer than the average person in Chidaville, they would be way more likely to not have access to a gym just because they cannot afford it. If many people in Chidaville don’t have access to internet, only the richest players data has been collected, and they are more likely to have access to the gym

Application of a Theoretical Probability • He estimates that if he rolls the dice 100 times, he will get a total of 6 16 or 17 times Do you agree with the conclusion above Assumed 6 outcomes, and I have never seen a dice not give an outcome of 1, 2, 3, 4, 5, 6 Assumed they are all equally likely: the dice came in a board game so I expect they have weighted their device well

SUMMARY True Probability Amost Deterministic Usually unknown, a concept which our estimates work towards Involves the running of an experiment/ the collection of data Experimental probability Probabilistic Theoretical probability Probabilistic Applying past to the future OR One Cross Section to Another Cross Section Applying a finding which was arrived at purely based on reasoning

Advantages and Disadvatages Advantages Experimental probability Applicable to a range of practical studies Disadvantages Need very large sample size to estimate probabilities which are either very close to zero, or very close to one. Takes a lot of time and effort to design a quality experiment/ properly collect the data. Doesn’t cost anything or take a lot of time Theoretical probability Little space to go wrong. Usually involves only 2 or 2 (often solid) assumptions Often only applicable to trivial situations

The main idea is that neither the modeled nor the experimental estimate of the TRUE probability is the true probability itself, rather a model of some sort.

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