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### Course: 7th grade (Illustrative Mathematics)>Unit 8

Lesson 2: Lesson 3: What are probabilities?

# Intro to theoretical probability

We give you an introduction to probability through the example of flipping a quarter and rolling a die. Created by Sal Khan.

## Want to join the conversation?

• what does p(h) mean?
• p(h) is the ORACLE's answer to your question about the future. Keep reading if you want to understand what does it mean :)

First, we want to understand what possibly can happen in our experiment. For example, if we roll a die, there are 6 different events that can happen:
{event-1, event-2, event-3, event-4, event-5, event-6}
We cannot favor any of these events, because we can end up with any side of the die being up.

If, for example, we win \$1,000,000 when the die shows 1 OR 5 on it's side, then suddenly event-1 and event-5 stop being for us an EVENT. The real event is winning \$1,000,000! This BIG event makes little event-1 and event-5 indistinguishable. So we reorganize our view on the structure of the die under the influence of that problem ― winning \$1,000,000.
Now we say, there is an event of WINNING {event-1, event-5} and event of LOSING {event-2, event-3, event-4, event-6}.

There is an ORACLE that knows what happend in the past and what will happen in the future. You tell him about your big event and he gives you a number. In this video this ORACLE is called p. You feed him our event WINNING and he should give some number from 0 to 1. In his moments of absolute certainty, when his third eye opens wide (that is, when he sees the future clearly), he gives us either certain 0 or a certain 1. When his third eye is half open and he wants to sleep, he will give you numbers in between 0 and 1.

Generally, the future is determined. You will either 100% WIN or 100% LOSE. The number that mighty ORACLE gives you depends on his understanding of the situation. If he totally loses the power to see the future and becomes a mere mortal like us, he will always give you the number 0.5 on every question you ask him. What is the probability of you to run across a dinosaur on the street? ;)
• At , if we assume that the coin can land on a corner and stand straight, will the probability become 1/3?
• I would assume that if you can stand the coin straight up, without flipping it, you would have a chance of landing it on it's corner, it would just be very unlikely. Therefore, it would not be 1/3, because it does not have the same probability as the heads/tails. I would assume that the probability of it landing straight up could be about as low as 11/100, just because the coin is very thin, and (assuming it is not a sphere) your faces of the coin are wider.
• 1. Why does a larger number of experiments bring the percentage of say getting heads closer to fifty percent?
2. How would you describe the probability of getting all tails in the flipping coin experiment?
• Is the Law of Large Numbers:

If you flip a coin #1 time you can have:
{[H] or [T]}
If you flip repet it 2 times you can have:
{[H,H],[H,T], or [T,H], [T,T]}
Now for the important part. If you don't care about the order you could say that the event [H,T] is equal to the event [T,H] so it'd be the same as:
{[H,H], 2[H,T], [T,T]}
the probability of each event would be:
P([H,H]) = 1/4
P([H,T]) = P([T,H]) = 2/4 =1/2
P([T,T]) = 1/4

So flip the coin 100 times and you would see that there are more combinations of HEADS & TAILS that add up to 50% each than any other.
• Alan, Beth Carlos and Diana were discussing their possible grades in mathematics class this grading period. Alan said, "If I get an A, then Beth will get an A." Beth said, "If I get an A, then Carlos will get an A." Carlos said, "If I get an A, then Diana will get an A." All of these statements were true, but only two of the students received an A. Which two received A's?
• Take this one person at a time.
If Alan gets an A, how many other people get an A?
If Beth gets an A, how many other people get an A?
If Carlos gets an A, how many other people get an A?
• What is the difference between Chance and Probability?
• Not much, really. They display the same statistic, but chance is formatted as a percentage while probability is a fraction. So chance would be displayed as, say 25%, while probability would be displayed as 1/4.
• At Sal says "How many equally likely possibilities are there". What is meant by equally likely possibility.
• Let me break this into two parts. First I'll talk about how many possibilities there are, then I'll talk about equally likely possibilities.

The number of possibilities is the number of different things that could happen in a given scenario. If you were flipping a coin, you could get either heads or tails, making two possibilities. Or if you were taking a test, you could get the following letter grades: A, B, C, D, or F. So there are five possibilities.

Equally likely possibilities means that all of the outcomes could happen with the same probability. Say it was a warm, sunny day, and I wanted to go swimming. The weather forecast shows these possibilities: 85% chance of no rain, 10% chance of rain, 5% chance of rain with thunderstorms. There are three possibilities in this scenario, but they are not equally likely possibilities. To have the outcomes be equally likely, they each have to happen just as often as each other. Coin flips have two equally likely possibilities because heads isn't more likely than tails, and tails isn't more likely than heads. Dice rolls are another example. No number on the die is more likely to be rolled than any other.

Have a good day! (:
• what does p(h) mean
• It basically means the probability (p) of the coin landing on heads (h).
• Are there any "fair coins" in real life? And if there are, can you give an example?
• No, a fair coin only exists in theory. Even if the sides were deemed perfectly flat and the perimeter perfectly circular, it's extremely unlikely that that would be true all the way down to the level of atoms.
• Theoretically, if I flipped a coin 1,000 times, it would land on heads 500 times and on tails 500 times. But that's just theoretical. What if I flipped a coin 1,000 times, and every single time it landed on heads? Is there a way to calculate the chances of that happening? (even if it is very small?)
• First, what's the probability of getting a head per flip? It is 1 / 2 = 0.5.

Now, what's the probability of getting 2 heads in 2 flip? You have to flip head for the first time, and for the second time as well, so it will be 0.5 * 0.5 = 0.25.

You can expand this to 1000 times, which is 0.5^1000.