Basic Probability

In Progress
Grade 3
Est. Time
60 mins
Related to Smart Teacher DB (Column)

What's the goal?

Help you understand probability from scratch!

Why does it matter?

A lot is uncertain. In order to plan for what will happen, we have to be able to guess what is likely to happen! Probability gives us the ways to figure that out.


Let's start with a normal coin, it has a head on one side and "tails" on the other.
If you were to flip this coin, you'll get heads half of the time and then tails about half of the time. We can write this like this:
  • Heads: 50%
  • Tails: 50%
But we also have a shorthand for this. If we call heads "H" and tails "T", we can say that P(H) = 50% and P(T) = 50%.
Now give it a try! Find a coin and flip it 10 times. Keep track of how many heads and tails you get.
Let me guess... you got 5 heads and 5 tails? Okay, you might not have, but this is the most likely answer. We can actually get the expected number of heads by doing the following math:
Probability of an Event * Number of Events = Expected Value
So, in this case:
Expected heads = 50% chance of heads * 10 coin flips = 5 expected heads
And for tails, we can do the same thing:
Expected tails = 50% chance of tails * 10 coin flips = 5 expected tails
But things don't always happen as expected! Let's say you got 6 heads and 4 tails... or 7 heads and 3 tails. Does that change anything about our math above?
Nope! It's just that you experienced a more rare or unlikely event. It's kind of rare that you'd flip 7 heads and 3 tails... but it will happen if you flip a coin enough times. You may even get 10 heads in a row! Later, we'll teach you how to calculate just how unlikely it is to flip 10 heads in a row.

But let's try a different exercise. Let's consider a normal six-sided die, with numbers 1 through 6 on each side. If we roll this die 20 times, how many sixes should we expect to get?
We can use the same formula as above! The probability of our event, rolling a six, is 1/6 or 18.3%. With 20 events:
Expected sixes = 1/6 chance of a six * 20 rolls of die = 20/6 = 3.33 sixes rolled
So we obviously can't roll 3.33 sixes. But if you give it a try, this does mean that 3 or 4 sixes are most likely.
As we'll learn later, we can actually calculate the exact probability that we will roll a six a certain number of times!

But for now let's try something that is common in board games: rolling two dice, and let's say we want to get the probability that we will roll a seven. How would we do that?
At first, the best thing we can do is list out all the possibilities for the two dice. This is called our sample space.
Because each dice has six potential values, let's say the first dice is a 1. Then the second dice could be 1, 2, 3, 4, 5, or 6: six different values. That's the same if the first dice is a 2, then 3, et cetera. So if you were to write them all out, you'd see that there are 6 possibilities for the first dice * 6 possibilities for the second dice = 36 total possible combinations.
That is good to know, and now we can figure out how many of those combinations add up to seven. For this, let's make a list:
  • If the first dice is a 1 and the second dice is a 6
  • If the first dice is a 2 and the second dice is a 5
  • If the first dice is a 3 and the second dice is a 4
  • If the first dice is a 4 and the second dice is a 3
  • If the first dice is a 5 and the second dice is a 2
  • If the first dice is a 6 and the second dice is a 1
So we have 6 different possibilities here to get a total of 7 on our dice. Thus, we can just say that the probability of rolling a seven or P(7) = 6/36 = 1/6 = 18.3%
Now, what about the chances that we roll an 11? Do you think this will be harder or easier? We get 11 if:
  • The first dice is a 5 and the second is a 6, or
  • The first dice is a 6 and the second is a 5
So we only have 2 ways to get 11... it's a good bit tougher! The probability of rolling an 11 or P(11) = 2/36 = 1/18 = 5.55%




What's Next?