Main content

### Course: Get ready for Precalculus > Unit 8

Lesson 1: Basic theoretical probability# Intuitive sense of probabilities

Think about what probabilities really mean. What does a probability of 0 mean? How about 1?

## Want to join the conversation?

- At6:37, how is .99999 repeating equal to one? It rounds to one, but how does that make it the same thing?(25 votes)
- Let's think about it. First, we have to understand that the 9's go on forever, so they don't just stop after a while. Now, can you think of any number that would fit between 0.9 repeating and one? 0.1? 0.001? Any of the numbers if added to 0.9 repeating would go over one. Therefore, there are no numbers that can be slipped between 0.9 repeating and one, and therefore the two numbers are the same.

We can also prove this algebraically.

Let`x = 0.999...`

(repeating)`10x = 9.999...`

`10x - x = 9.999... - 0.999...`

`9x = 9`

`9x/9 = 9/9`

Any number over itself (except zero) is one.`x = 1`

We just proved that 0.999... is equal to one. Another helpful thing to remember is that a number can have (at least) two decimal representations: 1 = 0.999...; 5 = 4.999... etc.(91 votes)

- my teacher forced me to do 7 days of this non-stop please help(27 votes)
- Say your cat ate ur computer, then ur science project ate ur cat(6 votes)

- I very much dislike this(26 votes)
- I 100% dislike this beat that(2 votes)

- Does probability help in real-life situations? If yes, how?(3 votes)
- Probability is the chance that basically anything can happen.

It is helpful in real-life situations because you can come up with the likeliness that it will or will not happen.

Probability is used for so many things. The weather is a good example. When you see the weather, it's all in probabilities. 20% chance for rain or a 50% chance of snow or a 10% chance of hail.

It's also very important in the stock markets, in gambling, in decision making, and many of the sciences!

An example:

// A doctor in an emergency room has to think very quickly when a patient comes in with a rare condition.

When he looks at the statistics, he determines there is a (1/10) chance the patient will survive unless he performs surgery. OR, a (9/10) chance the patient will die without surgery.

The surgery has a (9/10) chance of success, OR a (1/10) chance of failure.

The patient has the same chance of miraculously surviving without any medical intervention (which is 1/10) as they do with the surgery failing (which is 1/10), OR, they have a (9/10) chance of surviving if they allow the doctor to perform surgery.

Although this is very watered down and simplified, it's a good conceptual beginning. The actual variables and factors that can go into real-life scenarios is very complex and it can become very difficult very quick.(10 votes)

- I know this doesn't really have much of anything to do with probability but what are the chances of getting struck by lightning and having all those extra electrons trapped in your body coursing through your veins all in circles and rubbing together creating a lot of friction in your body that usually creates lightning in the sky but instead giving u lightning abilities that you have to adapt to controlling it with your cells in your body? Could someone please explain how that doesn't make sense or is impossible? I'm really curious how. Even though I know it sounds a bit stupid.(6 votes)
- Based on my understanding, a probability of 0 means "it's technically possible, but don't hold your breath expecting it to happen." Like a dart hitting the
**exact**center of a dartboard (an infinitesimally small point) or a dart hitting the**exact**edge of a dartboard (another infinitesimally small point).

But I'm not sure how to translate that understanding into the dice example. Could someone help clarify?(0 votes)- Well,if an event is technically possible, it means the event has a probability
*close to zero*, not exactly zero. It will be very close to zero, surely, but not exactly zero, which is a very important difference. The probability of a dice showing six 1000 times in a row or a dart hitting the exact center of a dartboard are events with*almost zero probability*but the probability of a dice showing 7 or a dart becoming invisible are events with**exactly**zero probability. If an event has zero probabilty, it is, technically or otherwise. Any event which is possible, no matter how unlikely it is, will have non-zero probability.**impossible**(16 votes)

- whats good yall(3 votes)
- At3:34it says if something is 0 then it is impossible, then does that mean if it is 0.00000001 it is impossible but still a tiny bit possible??(1 vote)
- If an event has probability 0.00000001, it is possible but extremely unlikely to happen.

Have a blessed, wonderful day!(10 votes)

- so if the probability of me getting an ice cream cone is 0, I wont get it?🍧🍨🍦❓❔❔❓(3 votes)
- Wahh.. Sad but true... no ice cream cone for you.(4 votes)

## Video transcript

- [Voiceover] What I
hope to do in this video is give ourselves a more
intuitive sense of probabilities. Let's go back to an example
that we've seen before. We're rolling a fair six-sided die. There are six possibilities. We could get a one, a two, a three, a four, a five or a six. Now let's say we ask ourselves what is the probability of rolling a number that is less than or equal to two? What is this going to be? Well, there are six equally
likely possibilities. Rolling less than or equal to two, well, that means I'm either
rolling a one or a two. So two, one, two, of the six equally likely possibilities
meet my constraints. So there is a 2/6 probability
of rolling a number less than or equal to two. Or I can just rewrite that
as an equivalent fraction. I could say there's a 1/3 probability. I could go either way. Now let's also ask
ourselves another question. What is the probability
of rolling a number greater than or equal to three? Once again, there are six
equally likely possibilities. How many of them involve rolling greater than or equal to three? Let's see, one, two, three, four, these possibilities right over here. Throw a three, a four, a five or six. So four out of the six
equally likely possibilities. Or I could rewrite this
as an equivalent fraction, as 2/3. So what's more likely? Rolling a number that's
less than or equal to two? Or rolling a number that's
greater than or equal to three? Well, you can see it right over here. The probability of rolling
greater than or equal to three is 2/3 while the probability
of rolling less than or equal to two is only 1/3. This number is greater. So this has a greater probability or another way of thinking about it, rolling greater than or equal to three is more likely than rolling
less than or equal to two. In fact, not only is it more likely, you see that 2/3 is twice 1/3. This right over here is twice as likely. You're twice as likely to
roll a number greater than or equal to three than
you are to roll a number less than or equal to two. You can even see right over here. You have twice as many possibilities of the six equally likely
ones, four versus two. Four versus two here. So you say, "Okay, I get it Sal." If the probability is a larger number, the event is more likely. It makes sense and in
this case, it's twice. The number is twice as large
so it's twice as likely. But what's the range of
possible probabilities? How low can a probability get and how high can a probability get? Let's think about the first question. How low can a probability go? How low, so what's the lowest probability that you can imagine for anything? Well, let's give ourselves a
little bit of an experiment. Let's ask ourselves the probability of rolling a seven. Well, once and pause the video and try to figure it out on your own. Well, there are six equally
likely possibilities. How many of them involve rolling a seven? Well, none of them. It's impossible to roll a seven. So none of the six. We could say this probability is zero. If you see a probability of zero, someone says the probability
of that thing happening is zero, that means it's impossible. That means in no world can that happen, if it's exactly zero. This right here, the probability is zero. That means it is impossible. It is impossible. Now how high can a probability get? How high can a probability get? Well, let's think about it. Let's say probability of rolling any number from one to six. Well, I have six equally likely possibilities
and any one of those six meets this constraint. I would have rolled a number, any number, from one to six, including one and six. So there are six equally
likely possibilities. So the probability is one. Someone says the probability
is zero, it's impossible. If someone says the probability is one, that means it's definitely
going to happen. It's definitely going to happen. So the maximum probability
for anything is one. The minimum probability is zero. You don't have negative probabilities and you don't have
probabilities greater than one. You might be thinking, "Wait, wait. "I've seen things that they look like "larger numbers than one." You're probably thinking of
seeing this as a percentage. One as a percentage,
you can also write this as 100%. This right over here as
a percentage is 100%. 100% is the same thing as one. You can't have a probability at 110%. 110% would be the same thing as 1.1. Now this is really interesting
because you'd often see someone say, "Hey, something
for sure is going to happen "or something is impossible." But even a lot of the things
that we think for sure are going to happen,
there's some probability or some chance that they don't happen. For example, you might hear someone say, "Well, what's the probability that the sun "will rise tomorrow?" Well, you might say it's
going to happen for sure. But you gotta remember some type of weird cosmological event might
occur, some kind of strange, huge planet-sized object
in space might come and knock the earth out of its rotation. Who knows what could happen? All these have a very low likelihood. Very, very, very, very,
very, very low likelihood. But it's hard to say it's exactly one. If I had said the probability that the sun will rise tomorrow, instead of saying one, I
would probably say it's 0.999. I would throw a lot of nines over here. I wouldn't say it's 0.9 repeating forever. Actually, there's an
interesting proof that 0.9 repeating forever is actually
the same thing as one, which is a little counterintuitive. But I would say there's
a very high probability. But even if it's such a high probability, it's going to be close to one. But I won't say it's exactly the one because there could be some
kind of quasar that blasts us with gamma rays. Or who knows what might happen? But it's a very, very high probability. Same thing, the probability here, probability that my pet gopher could write the next great novel. Writes a novel. Actually not just a novel, a great novel. Just a novel wouldn't be
that impressive for a gopher. Let's say great novel. Well, once again, this
gopher sitting there typing at a keyboard. It would seem somewhat random but there is some probability
that it actually does it. There's some chance it does it. So I would put this at a very low and I wouldn't say it's exactly zero. If we had an infinite number
of gophers doing this forever, who knows, maybe one of them
might write that great novel. In fact, if we had an infinite
number doing it forever, eventually, a lot of people would say, at some point you would. But just one gopher
trying to write a novel, what's the probability
they write a great novel? I would say it's pretty close to zero. I'd throw a lot of zeros here, and at some point you might
have something like this. Once again, not absolutely impossible but pretty close to, pretty, pretty close to impossible. So big takeaways? Higher probability, more likely. The lowest probability
you can get to? Zero. Highest probability is one. If your probability is more, when you're talking about coin flipping. If you say the probability
of heads for a fair coin and you say, "Well, that's 1/2," that means it's equally likely
to happen or not happen. Anything that has a larger
probability than 1/2, it's more likely to happen than not. Anything that has a
probability of less than 1/2, it's less likely to happen than not.