You are at a folk festival and want to draw lots. There are 100 tickets in the pot, 70 of which are losers. What is the probability that you pull a blank twice in a row?
This question cannot be answered directly. It makes sense to first bring the probabilities into a meaningful order. A useful tool for this is a tree diagram.
Tree diagram explanation
With a tree diagram you can bring order to a complex task.
A tree diagram is a way to visualize multi-stage random experiments. Each will Step treated individually so that the respective probabilities clear build on each other.
That is, in a tree diagram, you sort given events and probabilities so that you can more easily calculate the solution.
Create Tree Diagram – General Structure
As the name suggests, a tree diagram structured like a tree. With branches and ramifications beginning at a point, similar to the root of a tree. Since events and their relationships are represented visually in a tree diagram, it is also called a result tree designated.
Figure 1: General structure of a tree diagram
To create a tree diagram, do the following:
- There is one starting point, i.e. the root or the starting point from which as many branches branch off as there are events in the first stage of your random experiment. You write this at the end of the respective branch.
- The end of each branch represents a new branching point from which all possible events in the next stage branch out. After the 1st stage, for example, there are the 1st branching points. There you can enter the respective intermediate results. This works analogously for all other stages.
- The path from the beginning to one of the ends is called path designated. The end result is at the end of a path and therefore the end no longer represents a branching point.
- Each path describes an event of the entire multi-stage random experiment.
In Figure 1, the events denoted by A (or B) or (or ). The letters with the crossbar designate the respective counter-event, i.e. all events that do not contain A or B. This notation is commonly used when a random experiment has 2 possible outcomes.
You can also name your events differently. For example A and B or A1 and A2 and so on. This also applies if your random experiment has more than 2 possible outcomes. In this case, the event-counter-event procedure makes no sense anyway.
If you want to arrange the events of the lots mentioned above in a tree diagram, proceed as follows:
As a reminder: You are at a folk festival and want to draw lots. You fail 70% of the time. What is the probability that you fail again the second time?
- Step: You define all necessary events. It is up to you which designations you choose. For example, you can use for «nail» and for «no naughty» or N for «nail» and G for «win» or something else.
- Step: You pull out the necessary branches and write the events at the end of each one.
Figure 2: Tree diagram showing results for lots
On each level of the tree diagram you write down all the alternatives that can occur at that point in time.
You take a ticket. It can either be a fail or a win. So you note «N» in the 1st step and «G» on the parallel branch. Regardless of whether you lost or won, the second ticket can also be lost or a win. So you repeat the 1st step for both events.
You have now created your tree diagram. But how likely is it that after a smack, for example, you will get a smack again?
Probabilities in Tree Diagram – Path Rules
After you have put the events in a meaningful order, you can now add the probabilities. These are always entered on the corresponding branches.
Figure 3: Tree diagram with probabilities
The labels with the subscript letters – for example PA(B) – are conditional probabilities. But you can ignore them for now, that will be explained later.
In Figure 3, P(A) is the probability of event A, P(B) of event B, and so on. At the end of each path is the overall probability from the probabilities along the corresponding path. That is, for the path with probabilities P(A) and P(B), the total probability is P(A∩B).
The union ∩ denotes the intersection of two events. More specifically, it denotes the probability that A and B will occur together and at the same time.
Before we continue, here is a note:
the Sum of the probabilities of a level result together always 1because contain all possible outcomes of a random experiment, i.e. .
This allows you to calculate missing probabilities and check whether you have calculated correctly.
As a reminder, 70 of the 100 lots are rivets
In most cases, calculations are not based on absolute but on relative frequencies. To do this, you divide the number of desired events by the number of all events. For example, 70 rivets through every 100 lots gives a 0.7 chance of drawing a rivet.
So you enter the 0.7 on the branch to N. Since the probabilities of a stage always add up to 1, the probability of hitting a win is 0.3.
Figure 4: Tree diagram for lot probabilities
The same applies to the 2nd stage.
To circumvent the conditional probability mentioned earlier, this assumes that each ticket will be replaced before you draw the next ticket.
Now you can calculate the remaining probabilities. There are special calculation rules for tree diagrams.
1. Path rule
With the 1st path rule you can calculate the probability that A and B occur at the same time.
In a tree diagram you can for the probability of an event the probabilities along the associated path commonality multiply.
So if you want to calculate P(A∩B), you can multiply P(A) and P(B) together. So this rule only applies to a path and if you connect events with «and».
So if you want to calculate the probability of winning twice in a row, according to the 1st path rule, you multiply the probability of winning by the probability of winning (again).
It is further assumed that the lots are always directly replaced. So the probabilities always remain the same.
In the tree diagram, this corresponds to the path in turquoise.
Figure 5: Tree diagram for the 1st path rule
So you calculate:
And you get a as a result, which corresponds to converted.
Who will say a, need to say b as well. So there is also a 2nd path rule for the 1st path rule.
2. Path rule
You can use this if you want to calculate the probability that 2 (or more) paths occur.
In a tree diagram you can for the Probability of multiple events the probabilities of paths commonality add.
With the 2nd path rule you can, for example, calculate the probability that you will win at least 1 time if you draw 2 times.
In the tree diagram, this corresponds to all paths in which a win occurs, i.e. 3 out of 4 paths.
Figure 6: Tree diagram for the 2nd path rule
You calculate the probabilities of the individual paths using the 1st path rule
and you get the probability of at least one win with the 2nd path rule.
The probability of getting at least one win in 2 draws is , and , respectively.
So far, the number of lots was not taken into account, or it was the same for each train. However, how does the whole thing behave if the operator of the lottery booth does not replace the lottery tickets drawn?
Tree diagram without replacement
The case in which the number of events always remains the same or is irrelevant is called a in mathematics multi-level random experiment with replacementn. However, if the subjects become fewer in the course of the experiment, then it is a question of a multi-level random experiment without replacement.
You have to keep in mind that the number of events, as well as the total number of all possible events, changes after each round.
The stand at the folk festival has 100 lots, 70 of which are rivets. Every time you draw a ticket, the total number of tickets remaining and the number of fails or fails – depending on what you drew – decreases.
That is, if you draw 3 tickets in a row, each of which can be a fail or a win, after each draw you subtract 1 ticket from the total and a fail or a non-fail depending on what you drew.
Figure 7: Tree diagram without replacement
Relative frequency is used in the figure. You get this by dividing the number of events by the total number of all events. You can read more about this in the article «relative frequency».
This dependency of an event on the previous event conditional probability called.
If event B depends on event A, that is a conditional probability.
the conditional probability is the probability that an event occursafter a other event has already occurred is. The following applies to the calculation formula:
So called the conditional probability PA(B) the Probability of B under the Condition A.
Figure 8: Tree diagram with conditional probability
The ticket seller tries to motivate customers to buy a second ticket, claiming that if he replaces the tickets immediately, there is a good chance of getting the prize after a slump. Is he right?
You can check this by calculating the conditional probability for the 2nd lot using and comparing it with the initial values.
Figure 9: Lot conditional probabilities
You can use the conditional probability formula to calculate .
So it turns out that 30 of the 100 tickets are a win. Since the ticket seller replaces the tickets after each move, this is correct.
Here we deliberately calculated with replacement, since you also need the total probability for the calculation without replacement. That’s what we’re talking about below.
You also need the conditional probability if you want to flip a tree diagram.
Inverted tree diagram
To find out in a tree diagram what the overall probability of B, if A comes first you can flip it. It is then an inverse tree diagram.
in one inverse tree diagram will the events in your reversed order, so they are exactly the other way around. The final probabilities of each branch remain the same.
So if you have A in first place and B in second place in a tree diagram, B will be in first place and A in second place in the inverse tree diagram.
Figure 10: inverse…