Probability Tree Diagrams

Computing probabilities can be hard, sometimes nosotros add them, sometimes we multiply them, and often information technology is hard to figure out what to do ... tree diagrams to the rescue!

Here is a tree diagram for the toss of a money:

probability branch, probability and outcome

There are ii "branches" (Heads and Tails)

  • The probability of each branch is written on the co-operative
  • The effect is written at the end of the co-operative

Nosotros can extend the tree diagram to 2 tosses of a money:

probability tree 2 tosses of coin

How exercise we calculate the overall probabilities?

  • Nosotros multiply probabilities forth the branches
  • We add probabilities downwardly columns

probability tree calculations (multiply and add)

Now we can run across such things equally:

  • The probability of "Head, Head" is 0.5×0.5 = 0.25
  • All probabilities add to 1.0 (which is always a good check)
  • The probability of getting at least one Head from ii tosses is 0.25+0.25+0.25 = 0.75
  • ... and more

That was a elementary case using contained events (each toss of a coin is independent of the previous toss), but tree diagrams are actually wonderful for figuring out dependent events (where an event depends on what happens in the previous consequence) like this example:

soccer teams

Instance: Soccer Game

You are off to soccer, and dear being the Goalkeeper, but that depends who is the Coach today:

  • with Bus Sam the probability of being Goalkeeper is 0.5
  • with Motorbus Alex the probability of being Goalkeeper is 0.3

Sam is Coach more often ... almost half-dozen out of every 10 games (a probability of 0.6).

Then, what is the probability y'all volition be a Goalkeeper today?

Let's build the tree diagram. First we show the 2 possible coaches: Sam or Alex:

tree diagram 1

The probability of getting Sam is 0.6, so the probability of Alex must be 0.4 (together the probability is 1)

Now, if y'all get Sam, there is 0.5 probability of existence Goalie (and 0.5 of not beingness Goalie):

tree diagram 2

If you get Alex, there is 0.3 probability of existence Goalie (and 0.7 non):

tree diagram 3

The tree diagram is consummate, now let's summate the overall probabilities. This is done by multiplying each probability along the "branches" of the tree.

Here is how to do it for the "Sam, Yes" branch:

tree diagram 4

(When we take the 0.half-dozen chance of Sam being coach and include the 0.5 gamble that Sam will let you be Goalkeeper we end upwards with an 0.three chance.)

Only nosotros are not done however! Nosotros haven't included Alex as Coach:

tree diagram 5

An 0.4 chance of Alex every bit Jitney, followed by an 0.3 hazard gives 0.12.

Now we add the column:

0.3 + 0.12 = 0.42 probability of being a Goalkeeper today

(That is a 42% risk)

Bank check

I final step: complete the calculations and make sure they add to 1:

tree diagram 6

0.three + 0.3 + 0.12 + 0.28 = 1

Yep, it all adds up.

You can see more uses of tree diagrams on Conditional Probability.

Determination

Then there you go, when in doubtfulness draw a tree diagram, multiply along the branches and add together the columns. Make certain all probabilities add to 1 and y'all are practiced to go.