Find the probability of rolling

even numbers three times using a six-sided die

numbered from 1 to 6. So let’s just figure out the

probability of rolling it each of the times. So the probability of rolling

even numbers. So even roll on six-sided die. So let’s think about

that probability. Well, how many total

outcomes are there? How many possible rolls

could we get? Well, you get one, two, three,

four, five, six. And how many of them satisfy

these conditions, that it’s an even number? Well, it could be a 2,

it could be a 4, or it could be a 6. So the probability is the events

that match what you need, your condition for right

here, so three of the possible events are an even roll. And it’s out of a total of

six possible events. So there is a– 3 over 6 is

the same thing as 1/2 probability of rolling

even on each roll. Now they’re going to

roll– they want to roll even three times. And these are all going to

be independent events. Every time you roll, it’s not

going to affect what happens in the next roll, despite what

some gamblers might think. It has no impact on what happens

on the next roll. So the probability of rolling

even three times is equal to the probability of an even roll

one time, or even roll on six-sided die– this thing over

here is equal to that thing times that thing again. All right, that’s our first

roll– we copy and we paste it– times that thing and then

times that thing again. Right? That’s our first roll,

which is that. That’s our second roll. That’s our third roll. They’re independent events. So this is going to be equal to

1/2– that’s the same 1/2 right there– times 1/2

times 1/2, which is equal to 1 over 8. There’s a 1 in 8 possibility

that you roll even numbers on all three rolls. On this roll, this roll,

and that roll.

How do we prove the independence of 3 events that are not mutually independent using probability axioms and lemmas? @khanacademy

If you roll a d6 once the average roll is 3.5.

If you roll a d6 three times and take the middle die result, what's the average on that number? Is it still 3.5?

Example you roll 3d6 result is 1, 4 and 6, so the middle here is 4.

“despite what gamblers might think” ?