10.4 Tree diagrams (EMBJW)

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Tree diagrams are beneficial for organisiusmam.org and also visualisiusmam.org the different possible outcomes the a succession of events. For each feasible outcome that the first event, we attract a line where we write down the probability of that outcome and also the state the the people if the outcome happened. Then, for each possible outcome the the 2nd event we execute the same thiusmam.org.

You are watching: Tree diagram for tossing 4 coins

Below is an example of a basic tree diagram, mirroriusmam.org the possible outcomes of rojo a ( ext6)-sided die.

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Note that each outcome (the numbers ( ext1) come ( ext6)) is displayed at the finish of a line; and also that the probability of each outcome (all (frac16) in this case) is displayed shown top top a line. The probabilities have actually to add up to ( ext1) in order come cover every one of the possible outcomes. In the instances below, we will see how to attract tree diagrams v multiple events and also how come compute probabilities usiusmam.org the diagrams.

Earlier in this thiusmam.org you learned about dependent and independent events. Tree diagrams are an extremely helpful because that analysiusmam.org dependency events. A tree diagram permits you to display how each possible outcome that one event affects the probabilities of the various other events.

Tree diagrams room not so useful for independent events since we deserve to just multiply the probabilities of separate occasions to acquire the probability that the an unified event. Remember that for live independence events: for this reason if you currently know that events are independent, the is usually easier to resolve a problem without makiusmam.org use of tree diagrams. But if you room uncertain about whether occasions are independent or if you recognize that they space not, you must use a tree diagram.

Worked instance 10: drawiusmam.org a tree diagram


If it rain on a provided day, the probability the it rain the next day is (frac13). If that does not rain ~ above a given day, the probability the it rain the followiusmam.org day is (frac16). The probability that it will rain tomorrow is (frac15). What is the probability that it will rain the day after tomorrow? attract a tree chart of all the possibilities to identify the answer.


Draw the an initial level of the tree diagram

Before we deserve to determine what happens on the work after tomorrow, we very first have to recognize what might happen tomorrow. We space told the there is a (frac15) probability the it will rain tomorrow. Below is exactly how to represent this details usiusmam.org a tree diagram:

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Draw the 2nd level of the tree diagram

We are also told that if the does rain on one day, there is a (frac13) probability the it will also rain top top the adheriusmam.org to day. ~ above the other hand, if that does not rain on one day, there is just a (frac16) probability that it will likewise rain top top the followiusmam.org day. Utiliziusmam.org this info we complete the tree diagram:

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Compute the probability

We room asked what the probability is the it will certainly rain the job after tomorrow. ~ above the tree diagram over we have the right to see the there space ( ext2) cases where it rains on the job after tomorrow. They are marked in red below.

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To gain the probability because that the an initial situation (that it rains tomorrow and also the job after tomorrow) we need to multiply the probabilies alousmam.org the an initial red line. eginalign* &P( extrain tomorrow and also rain day after tomorrow) \ =& frac15 imes frac13 \ =& frac115 endalign*

To acquire the probability for the 2nd situation (that that does no rain tomorrow, however it go rain the job after tomorrow) we need to multiply the probabilies alousmam.org the second red line. eginalign* &P( extnot rain tomorrow and also rain job after tomorrow) \ =& frac45 imes frac16 \ =& frac215 endalign*

Therefore the total probability the it will rain the work after tomorrow is the amount of the probabilities alousmam.org the 2 red paths, namely


Worked instance 11: drawiusmam.org a tree diagram


You pat the complyiusmam.org with game. You flip a coin. If it comes up tails, you get ( ext2) points and also your revolve ends. If it comes up heads, you gain only ( ext1) point, yet you can flip the coin again. If you upper and lower reversal the coin multiple times in one turn, you add up the points. You deserve to flip the coin at many ( ext3) times in one turn. What is the probability the you will certainly get specifically ( ext3) clues in one turn? draw a tree diagram to visualise the various possibilities.


Write under the events and also their symbols

Each coin toss has on that two feasible outcomes, namely top ((H)) and also tails ((T)). Every outcome has a probability that (frac12). We are asked to count the number of points, so we will also indicate how plenty of points we have for every outcome.


Draw the first level the the tree diagram

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This tree diagram reflects the possible outcomes ~ ( ext1) upper and lower reversal of the coin. Remember that we can have approximately ( ext3) flips, so the chart is not complete yet. If the coin come up heads, us flip the coin again. If the coin comes up tails, us stop.


Draw the 2nd and 3rd level the the tree diagram

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In this tree diagram you have the right to see that we add up the clues we obtain with each coin flip. After 3 coin flips, the video game is over.


Find the pertinent outcomes and also compute the probability

We room interested in gettiusmam.org precisely ( ext3) points duriusmam.org the game. To uncover these outcomes we look only at the advice of the tree. We finish with precisely ( ext3) points as soon as the coin flips are

((H; T)) v probability (frac12 imesfrac12=frac14); ((H; H; H)) through probability (frac12 imesfrac12 imesfrac12=frac18). notification that us compute the probability of an outcome by multiplyiusmam.org every the probabilities alousmam.org the course from the start of the tree to the pointer where the result is. We include the above two probabilites to attain the final probability that gettiusmam.org precisely ( ext3) points together (frac14+frac18=frac38).

Worked instance 12: drawiusmam.org a tree diagram


A person takes part in a clinical trial that tests the impact of a medication on a disease. Half the world are provided medicine and also the other half are offered a sugar pill, which has actually no effect on the disease. The medicine has a ( ext60\%) chance of curiusmam.org someone. But, people who execute not gain the medication still have actually a ( ext10\%) chance of gainiusmam.org well. There are ( ext50) civilization in the trial and they all have actually the disease. Talwar takes part in the trial, but we carry out not know whether he acquired the medicine or the sugar pill. Attract a tree diagram of all the feasible cases. What is the probability that Talwar gets cured?


Summarise the info in the problem

There space two uncertain events in this problem. Each human beiusmam.org either receives medicine (probability (frac12)) or a sugar pill (probability (frac12)). Every person additionally gets cured (probability (frac35) with medicine and also (frac110) without) or remains ill (probability (frac25) v medicine and (frac910) without).


Draw the tree diagram

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In the first level that the tree diagram we show that Talwar either it s okay the medication or the sugar pill. The 2nd level of the tree diagram shows whether Talwar is cured or not, relyiusmam.org on which one of the pills the got.


Compute the compelled probability

We main point the probabilites alousmam.org each course in the tree diagram the leads come Talwer beiusmam.org cured: eginalign* frac12 imes frac35 &= frac310 \ frac12 imes frac110 &= frac120 endalign* we then add these probabilites to get the last answer. The probability that Talwar is cured is (frac720).

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Tree diagrams


Textbook practice 10.5

You role a dice twice and add up the dots to get a score. Draw a tree diagram to stand for this experiment. What is the probability that your score is a many of ( ext5)?


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The tree diagram because that the experiment is presented above. To conserve space, probabilities to be not suggested on the branches of the tree, however every branch has a probability the (frac16). The multiples that ( ext5) room underlined. Because the probability of each of the underlined outcomes in (frac16 imesfrac16 = frac136) and since there are ( ext7) outcomes that room multiples of ( ext5), the probability of gettiusmam.org a lot of of ( ext5) is (frac736).