## Ask Professor Puzzler

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Vusi from South Africa wants to know - if you roll a blue die and a yellow die, why are these considered to be independent events?

Well, Vusi, let's start off by making sure we understand the terms "dependent events" and "independent events."

**Dependent Events**

Two or more events are dependent if the outcome of one of the events will affect the outcome of the other.

For example, if you have a jar that contains 100 blue jelly beans and 100 yellow jelly beans, and you draw one jelly bean and eat it. The probability that the jelly bean will be blue is 0.5 (100/200, because there are 100 blue jelly beans and a total of 200 jelly beans all together). However, if you then draw a second jelly bean from the jar, the probability of it being blue is *not* 0.5. In fact, we really don't *know *what the probability is. Why? Because if the first draw was a blue, the probability will be 99/199 (because we have one less blue, and one less total), but if the first draw was yellow, the probability of a blue on the second draw will be 100/199 (because we still have 100 blues, but one fewer yellow). Thus, the probability of the second event can't be calculated without knowing the result of the first event. The first event affects the outcome of the second event. These are therefore *dependent events*.

**Independent Events**

Two or more events are independent if the outcome of one event has no effect on the outcome of another.

To keep this simple, let's talk about the same jar. Only this time, instead of eating the jelly bean, you put it back in the jar after drawing it out. The probability that the first jelly bean drawn is blue is 0.5. But what about the second drawing? In the second drawing, you still have 100 blue jelly beans and a total of 200, because the first jelly bean drawn was put back in. Thus, it makes no difference what you draw the first time; the probability for the second draw being blue is still 0.5. The first draw does not affect the second draw, so these are *independent events.*

Hopefully, with this in mind, you can answer your own question. Are the rolls of a blue die and a yellow die independent? Yes they are! Because the roll of the blue die is not going to affect the roll of the yellow die. The yellow die isn't going to think, "Oh, the blue die is showing a 3, so I better not land on 3," or "The blue die is showing an even number, so I should show an odd," or "Hey, the blue die is showing a 6, so if I land on 6 we'll have doubles!" Whatever happened with the blue die has absolutely no bearing on what happens with the yellow die. Therefore, these are independent events.

By the way, we have a reference unit on probability on this site, which you can find here: Probability concepts and problems. This unit contains a section on independent events and dependent events, so you can read more on that here: Independent Events, Dependent Events.

Thanks for asking!