The AND and OR of probability (2 Viewers)

[oasfree]

I wonder if you guys at HS ever try to work out the rule for AND and OR when it comes to working out probability?

For example, if you toss a coin, the probability of a head is 1/2. If you toss it again, the probability of a head is again 1/2. Therefore the possibility of tossing two heads in two tosses is 1/2 x 1/2 = 1/4. I know I was taught how to work this out a long time ago. But I did not remember that teachers actually prove to me that it was the correct way to time the two probabilities together.

Similarly, the OR rule is to plus the two together. But I also do not remember teachers actually showed me why it worked like that.

Did your teachers bother to prove that these rules work? Or some how show you to derive the rules yourself?

It bugged me for a long time until I tried to work some things out myself much later when I started to doubt the rules. There is good reason to doubt when the teaching was so superficial. For example, they said "the probability is 1/2 to get a head in a toss". Fair enough. Then they said again "If you now toss again, the probability is still 1/2 to get a head". Again, that's fair enough. So I would assume that if I toss again, the probably to get a head is still 1/2 and so on. But then they also said "If you toss many times, the number of heads and tails will be pretty even". And the teaching stopped here! It's so easy to get to the wrong conclusion that after I get 2 heads from 2 tosses, the next toss has a higher probability of being a tail than 1/2. That leads to confusion. The trouble is that I have never had time or will to revisit the issue myself and sort all the gaps in my knowledge about chance out completely. I wish they taught me a bit more.

Any one out there feel this way? Honestly I feel I am much better in math than half of the math teachers out there. So I guess they probably never bother to go deep nowadays.

 

[LordPc]

I wonder if you guys at HS ever try to work out the rule for AND and OR when it comes to working out probability?

For example, if you toss a coin, the probability of a head is 1/2. If you toss it again, the probability of a head is again 1/2. Therefore the possibility of tossing two heads in two tosses is 1/2 x 1/2 = 1/4. I know I was taught how to work this out a long time ago. But I did not remember that teachers actually prove to me that it was the correct way to time the two probabilities together.

Similarly, the OR rule is to plus the two together. But I also do not remember teachers actually showed me why it worked like that.

Did your teachers bother to prove that these rules work? Or some how show you to derive the rules yourself?

It bugged me for a long time until I tried to work some things out myself much later when I started to doubt the rules. There is good reason to doubt when the teaching was so superficial. For example, they said "the probability is 1/2 to get a head in a toss". Fair enough. Then they said again "If you now toss again, the probability is still 1/2 to get a head". Again, that's fair enough. So I would assume that if I toss again, the probably to get a head is still 1/2 and so on. But then they also said "If you toss many times, the number of heads and tails will be pretty even". And the teaching stopped here!

It's so easy to get to the wrong conclusion that after I get 2 heads from 2 tosses, the next toss has a higher probability of being a tail than 1/2. That leads to confusion. The trouble is that I have never had time or will to revisit the issue myself and sort all the gaps in my knowledge about chance out completely. I wish they taught me a bit more.

Any one out there feel this way? Honestly I feel I am much better in math than half of the math teachers out there. So I guess they probably never bother to go deep nowadays.

I understand what your saying but dont feel too bad about it. My teacher would prove to us each formula that we used so we understood how we got there. Then she would say something like "So now you know that the formula is correct and works, so now you can use it". Point is that you dont need to know how to prove most formulae, you just need to know they work and be able to use them. So its not all that important in truth.

There are some rules that you actually do need to know how to prove though, but I think this is mainly for 3u and 4u topics and even then is rare.

and the heads and tails thing is always confusing, especially because of wording. Chance is always 50/50 but you must understand that while chances of heads is always 50%, chance of consecutive heads (ie, H, T, H, H, H, H, H, H, etc) is quite low. People often mistake as the chance of getting tails rises as you get more heads.

 

[oasfree]

I will try to solve some of the important basic gaps I still have to see what my teachers have missed out. The basics of probability is very shaky! Say if you have 1 coin, you toss it 3 times. They say probability that you get 3 heads is 1/8. Is this valid? It's very hazy here.

I would say that if you have 3 coins, and you toss 1 time for all three of them, you get 1/8. It's simple as we just look at the 8 total combinations to see there is only 1 set of 3 heads. But by having one coin and tossing 3 times, it's a different animal altogether. As the observation that you tend to get 50% of heads and 50% of tail over a reasonable number of tosses, it's clear that after a string of heads you will get more tails. So it suggests that the probability that the next toss will give a tail is higher than a head. But this is direct contradiction with the idea that every toss is a single act and should give 1/2 probability to get head or tail. There is a gap in the understanding here. The act of combining probabilities in human mind over many tosses seem like more of "mind over matter" rather than pure physics. We all know that if you see some one losing heavily on a betting machine, you should wait and play right after that. The probability of winning is much higher after that is because the machine is set to pay out about 70% of all what it takes in. So at some time it will start to pay out to make good of this programming.

Similarly in a game at casino, if you play head and tail. After seeing too many heads you would play tail because the number of heads and tails must even out over about 50-100 throws. It looks undeniably that some throws have more possibility of landing a head or landing a tail. The probability does not look static at 50% at each throw. But how could this be possible without a contradiction?

I will have to consult some math specialist and do some calculations myself to understand more about this gap of knowledge.

 

[bored of sc]

I haven't done probability yet but I suggest using the principle AND = multiply, OR = add.

 

[oasfree]

I haven't done probability yet but I suggest using the principle AND = multiply, OR = add.

But this is precisely the problem. Teachers give you the rule that was copied from a book and you apply it to get a result. Then teachers and you pray to God that the rule is correct. It feels like you start from some where you don't know and end up in somewhere you don't know.

 

[Trebla]

The intuition behind addition is to sum all possible outcomes. So when you have outcome A or outcome B which do not occur together, they are added because you want the total proportion of outcome A along with outcome B in your total.
A good analogy is a pie chart where a fraction is coloured red and another fraction is coloured blue. The probability of getting either red or blue is the sum of the fractions of the chart that make up each of them.

With multiplication, this is done when outcomes occur 'together'. So when you have outcome A and outcome B occuring together independently, they are multiplied because outcome A can pair with outcome B with the probability outcome B occurs.
So taking the coin tossed twice example, the probability a head occurs is 1/2. When you toss the coin a second time, this head can pair up with either another head or a tail. There is a 1 in 2 chance the first head can pair up with another head. So the probability of 2 heads is 1/2 x 1/2 where the first 1/2 is the original probability of a head and the second half is the probability this head pairs up with another head (I think this is called conditional probability). If you draw a 2 x 2 table of all possible outcomes HT, TH, HH or TT, then the probability HH occurs is 1/4.

It's so easy to get to the wrong conclusion that after I get 2 heads from 2 tosses, the next toss has a higher probability of being a tail than 1/2.

This is where the concept of independence arises. The next toss is unaffected by the previous toss, so the probability remains the same.
The idea of convergence is due to the central limit theorem which states that in theory the number of heads and tails should roughly equalise because it has a high probability of doing so(according to standard normal distribution). It's a theoretical concept that simply means it is highly unlikely in 20 tosses to get say 15 heads and 5 tails (perhaps evidence of some bias towards heads).

 

[Templar]

In the case of poker machines and casino gambling, you also have regression towards the mean.

 

[oasfree]

I prompted myself to re-read the theory of probability that is in 1st year University mathematics. I see now that probability is completely linked to statistics. The concept itself is about using fraction to describe the chance in a best possible way. As the events are repeated indefinitely, the values become more reasonable (you get about same number of Hs and Ts assuming using fair coins).

I then realise it's way beyond HS level to go very deep. Some philosophers who believe in determinism would not accept chance at all. They believe that the reason why a coin lands H because it was tossed in certain way and the physical forces in the environment cause the landing to be H. But the math of probability still works as you toss many times, each time the forces are different so eventually you reach about same amount of Hs and Ts. In this light, the toss of the next coin with the assumption that the probability of getting H or T is at precisely 50%, is in fact invalid. But it does not matter as in the long run after a lot of attempts over time, the distribution will average out at about 50%.

I suppose I just have to accept that at HS level, AND = multiply is the only way to teach provided that kids are warned the result is not as "trustworthy" as 2 x 2 = 4. This is especially true if you are in a casino.

This links back to the root of statistics as a method to measure what otherwise impossible to measure because of all the unknown forces. Therefore statistics is humourously also called the art of lying.

 

[lolokay]

what's wrong with "AND = multiply"? I thought that was pretty intuitive

say, you're rolling a dice and if you don't roll a 6 you stop rolling
there's a 1/6 chance of rolling 6 on the first roll
if you made it to the second roll, there would be a 1/6 chance again of rolling 6
so the chance of rolling two sixes is 1/6 the chance of rolling the first six, = 1/6 * 1/6
etc

 

[oasfree]

what's wrong with "AND = multiply"? I thought that was pretty intuitive

say, you're rolling a dice and if you don't roll a 6 you stop rolling
there's a 1/6 chance of rolling 6 on the first roll
if you made it to the second roll, there would be a 1/6 chance again of rolling 6
so the chance of rolling two sixes is 1/6 the chance of rolling the first six, = 1/6 * 1/6
etc

If you get back to my original post, there is a possibility of contradiction. Once you are "lucky" to get a bunch of Hs in a row, it's hard to imagine that the chance is exactly 1/2 that you will get another H in your next toss because another rule suggests that the number of Hs and Ts will be likely to be similar after a while. Therefore, if you are in a casino and you have seen a string of Hs, you would bet T knowing soon the numbers of Hs and Ts will be similar. So more Ts would come out later when you reach to about 50 tosses. If the chance is not exactly then the rule "AND = multiply" won't produce exact chance either.

So technically, it's not correct to answer questions like "What is the chance of getting 2 H in 2 tosses" as you would say 1/2 x 1/2 = 1/4. But it's correct, if you add "provided that you keep doing this tossing act over 100 times". This is the bit teachers never seem to bother to tell students.

Also I don't like the fact that many people can try to use the AND rule indiscriminately. It is supposed that you can use it to find the combined chance of independent events. It's good for the case of tossing coins. Two tosses are independent. But when they use it to connect two independent events of completely different nature, I fear it makes no sense at all.

A - The chance the US loses the War in Iraq, say 9/10
B- The chance that women have mustache, say 1/10

A and B = 9/10 x 1/10 = 9/100

That does not make any sense. I have not encounter any guidelines that probability has to be employed within a meaningful sample space. Perhaps my teachers all the way from HS to University did not bother to tell me about how to use probability properly. I have seen questions that use AND to connect events that don't seem to be in same sample space.

 

[lolokay]

there's still a 1/2 chance of getting H. it's just very unlikely that you got the string of H's in the first place

as for your weird example, if there is a 9/10 chance that "the US loses the War in Iraq" and a 1/10 chance that "women have mustache", and they're independent events, then there would indeed be a 9/100 that both of these events occur

still not sure what you're saying

 

[Trebla]

If you get back to my original post, there is a possibility of contradiction. Once you are "lucky" to get a bunch of Hs in a row, it's hard to imagine that the chance is exactly 1/2 that you will get another H in your next toss because another rule suggests that the number of Hs and Ts will be likely to be similar after a while. Therefore, if you are in a casino and you have seen a string of Hs, you would bet T knowing soon the numbers of Hs and Ts will be similar. So more Ts would come out later when you reach to about 50 tosses. If the chance is not exactly then the rule "AND = multiply" won't produce exact chance either.
So technically, it's not correct to answer questions like "What is the chance of getting 2 H in 2 tosses" as you would say 1/2 x 1/2 = 1/4. But it's correct, if you add "provided that you keep doing this tossing act over 100 times". This is the bit teachers never seem to bother to tell students.

You are interpreting it in a way such that there dependence on previous outcomes, in which case the 'probability' will of course 'differ' when taking dependence into account. You are meant to interpret the outcomes as independent of each other in which case the multiplication applies.

Also, statistical inference can tell you that if there is a highly unlikely event that you get many heads in a row, then there is a strong probability that the coin is biased and a very low probability it is fair. In technical terms, the probability of achieving its expected value E(X) where X is the number of heads may not be 0.5 (i.e. if you get too many heads there is a high probability that the probability of getting heads is higher than 0.5). Confused? Well such a concept would be very difficult to comprehend by the average high school student.

Also, the probability model for the coin tossing scenario is the discrete binomial model. In high school, it is assumed that the coin tossing scenario follows a discrete binomial distribution and we work in terms of expected values to keep it simple. We assume it follows the expected values of the binomial model so that we don't need to confuse people with the central limit theorem (i.e. we don't have to say it has a probability of 0.5 only with several tosses). Should this model come under question then we would have to attempt to fit another probability model in order to predict outcomes.

Keep in mind probability models are not exactly descriptive of reality but are theoretical to give a good approximation. So should there be an unusually high number of heads, we can say that under the binomial model this is an extremely unlikely event (since under this model the probability of achieving a head in each toss is 0.5) but under another model like say the Poisson model, this may not be so unlikely. This is basically the idea of statistical inference which is covered in university statistics. The very concept of probability models itself is generally difficult for the average student to grasp.

Also I don't like the fact that many people can try to use the AND rule indiscriminately. It is supposed that you can use it to find the combined chance of independent events. It's good for the case of tossing coins. Two tosses are independent. But when they use it to connect two independent events of completely different nature, I fear it makes no sense at all.

A - The chance the US loses the War in Iraq, say 9/10
B- The chance that women have mustache, say 1/10

A and B = 9/10 x 1/10 = 9/100

That does not make any sense. I have not encounter any guidelines that probability has to be employed within a meaningful sample space. Perhaps my teachers all the way from HS to University did not bother to tell me about how to use probability properly. I have seen questions that use AND to connect events that don't seem to be in same sample space.

There is nothing wrong with what you did there. All it says is that the probability A occurs and B occurs at the same time independently is 9/100. The events don't have to be similar or be in the same sample space at all. All that number means is that there is a 9% chance that the event of the US losing the war and the event of a woman having a moustache occur together.