ShmoopTube

00:03

Here's a shmoopy question for you...

00:05

When a normal Vegas pair of dice with six sides is tossed, what is the probability that

00:11

the sum of the numbers will be greater than or equal to 3?

00:15

You've already gotten in trouble once for sneaking in a pair of your weighted dice,

00:19

so we know you wouldn't dare pulling that stunt again.

00:22

And here are the potential answers...

00:27

OK, so what is this question really asking?

00:30

Well, it's a two-parter -- that is, 2 events happen and we have to figure out the odds

00:34

of the number from roll 1 plus the number from roll 2 being greater than or equal to 3.

00:40

If we just think about this a moment before doing ANY math, the SUM of these

00:45

rolls is almost ALWAYS going to be greater than 3 -- like,

00:49

the worst we could do is a 1 and a 1 but how many times does that happen?

00:53

If we rolled a 2 and a 1, we'd be at least equal to three -- same deal with a 1 and a 2.

00:59

Then EVERY OTHER ROLL combo is greater than or equal to 3.

01:03

So we KNOW that the odds are going to be VERY

01:06

high that we solve the question with a yes... or a true.

01:09

And if we just glance at the answer set, we could probably very quickly throw out answers

01:13

A and B because those show long odds and ours are going to be... the opposite.

01:19

If we had to, we could brute force with this question by laying out all of the 36 combinations

01:24

of rolls... like this...

01:25

...and counting how many numbers are greater than or equal to 3.

01:29

But that's a lotta work and time consuming and doesn't scale -- like... what if they

01:32

asked us for 5 dice rolls' odds? Yeah, then we'd be up test creek without a pencil.

01:39

So instead, since we know that the odds are going to very high, it'd be easier for us

01:44

to count the number of times the sum of dice is less than 3, or equal to 2, since 1 + 1

01:49

= 2, and 1 is the lowest number on a dice. We also know that 6 times 6, or 36, tells

01:55

us how many possible sum combinations we can get.

01:59

Not how many unique sums, but just

02:01

how many combinations we could get in total if we really wanted to...

02:05

Now we can apply the principle that the probability

02:08

of something occurring is the same as 1 minus the probability of that something not occurring.

02:14

In math speak, this would be P = 1 - (not P).

02:17

In this case, the 'not P' is the probability that the sum of the dice, when rolled,

02:22

is 1 over 36, which we found just a few seconds ago.

02:26

So, P = 1 - 1/36 equals 35/36.

02:31

The correct answer is D.

02:34

And yet... cat's eyes. Interesting.

02:36

You might not want to keep those dice of yours in your back pocket...

Full Transcript