Multiplication rule for independent events
Die rolling probability with independent events
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.