Multistage Experiments with Tree Diagrams

Calculating probabilities can be hard, sometimes you add them, sometimes you multiply them, and often it is hard to figure out what to do ... tree diagrams to the rescue!

You can use Tree Diagrams for either an Independent Events or Dependent Events.

Independent Event: When the occurrence of one event has no influence on the outcome of a second event. An example of 2 independent events are; say you rolled a die and flipped a coin. The probability of getting and number face on the die in no way influences the probability of getting a head or a tail on the coin.

Dependent Event: The outcome of the event affects the second event. I found a great example of this on  wyzant 

For example, if you were to draw a two cards from a deck of 52 cards. If on your first draw you had an ace and you put that aside, the probability of drawing an ace on the second draw is greatly changed because you drew an ace the first time. Let's calculate these different probabilities to see what's going on.

There are 4 Aces in a deck of 52 cards

On your first draw, the probability of getting an ace is given by:

If we don't return this card into the deck, the probability of drawing an ace on the second pick is given by

As you can clearly see, the above two probabilities are different, so we say that the two events are dependent. The likelihood of the second event depends on what happens in the first event.

Here is a great video of showing how to use a tree diagram for finding the Probability of 4 different situations.

It really helps having a visual of what you are trying to find instead of just doing the work in your head.