3.5 Tree and venn diagrams

Introductory statistics Page 1 / 10

Sometimes, when the probability problems are complex, it can be helpful to graph the situation. Tree diagrams and Venn diagrams are two tools that can be used to visualize and solve conditional probabilities.

Tree diagrams

A tree diagram is a special type of graph used to determine the outcomes of an experiment. It consists of "branches" that are labeled with either frequencies or probabilities. Tree diagrams can make some probability problems easier to visualize and solve. The following example illustrates how to use a tree diagram.

In an urn, there are 11 balls. Three balls are red ( R ) and eight balls are blue ( B ). Draw two balls, one at a time, with replacement . "With replacement" means that you put the first ball back in the urn before you select the second ball. The tree diagram using frequencies that show all the possible outcomes follows.

This is a tree diagram with branches showing frequencies of each draw. The first branch shows two lines: 8B and 3R. The second branch has a set of two lines (8B and 3R) for each line of the first branch. Multiply along each line to find 64BB, 24BR, 24RB, and 9RR. — Total = 64 + 24 + 24 + 9 = 121

The first set of branches represents the first draw. The second set of branches represents the second draw. Each of the outcomes is distinct. In fact, we can list each red ball as R 1, R 2, and R 3 and each blue ball as B 1, B 2, B 3, B 4, B 5, B 6, B 7, and B 8. Then the nine RR outcomes can be written as:

R 1 R 1
R 1 R 2
R 1 R 3
R 2 R 1
R 2 R 2
R 2 R 3
R 3 R 1
R 3 R 2
R 3 R 3

The other outcomes are similar.

There are a total of 11 balls in the urn. Draw two balls, one at a time, with replacement. There are 11(11) = 121 outcomes, the size of the sample space .

a. List the 24 BR outcomes: B 1 R 1, B 1 R 2, B 1 R 3, ...

B 1 R 1
B 1 R 2
B 1 R 3
B 2 R 1
B 2 R 2
B 2 R 3
B 3 R 1
B 3 R 2
B 3 R 3
B 4 R 1
B 4 R 2
B 4 R 3
B 5 R 1
B 5 R 2
B 5 R 3
B 6 R 1
B 6 R 2
B 6 R 3
B 7 R 1
B 7 R 2
B 7 R 3
B 8 R 1
B 8 R 2
B 8 R 3

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b. Using the tree diagram, calculate P ( RR ).

b. P ( RR ) = $(\frac{3}{11}) (\frac{3}{11})$ = $\frac{9}{121}$

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c. Using the tree diagram, calculate P ( RB OR BR ).

c. P ( RB OR BR ) = $(\frac{3}{11}) (\frac{8}{11})$ + $(\frac{8}{11}) (\frac{3}{11})$ = $\frac{48}{121}$

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d. Using the tree diagram, calculate P ( R on 1st draw AND B on 2nd draw).

d. P ( R on 1st draw AND B on 2nd draw) = P ( RB ) = $(\frac{3}{11}) (\frac{8}{11})$ = $\frac{24}{121}$

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e. Using the tree diagram, calculate P ( R on 2nd draw GIVEN B on 1st draw).

e. P ( R on 2nd draw GIVEN B on 1st draw) = P ( R on 2nd| B on 1st) = $\frac{24}{88}$ = $\frac{3}{11}$

This problem is a conditional one. The sample space has been reduced to those outcomes that already have a blue on the first draw. There are 24 + 64 = 88 possible outcomes (24 BR and 64 BB ). Twenty-four of the 88 possible outcomes are BR . $\frac{24}{88}$ = $\frac{3}{11}$ .

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f. Using the tree diagram, calculate P ( BB ).

f. P ( BB ) = $\frac{64}{121}$

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g. Using the tree diagram, calculate P ( B on the 2nd draw given R on the first draw).

g. P ( B on 2nd draw| R on 1st draw) = $\frac{8}{11}$

There are 9 + 24 outcomes that have R on the first draw (9 RR and 24 RB ). The sample space is then 9 + 24 = 33. 24 of the 33 outcomes have B on the second draw. The probability is then $\frac{24}{33}$ .

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Try it

In a standard deck, there are 52 cards. 12 cards are face cards (event F ) and 40 cards are not face cards (event N ). Draw two cards, one at a time, with replacement. All possible outcomes are shown in the tree diagram as frequencies. Using the tree diagram, calculate P ( FF ).

This is a tree diagram with branches showing frequencies of each draw. The first branch shows two lines: 12F and 40N. The second branch has a set of two lines (12F and 40N) for each line of the first branch. Multiply along each line to find 144FF, 480FN, 480NF, and 1,600NN.

Total number of outcomes is 144 + 480 + 480 + 1600 = 2,704.

P ( FF ) = $\frac{144}{144 + 480 + 480 + 1,600} = \frac{144}{2, 704} = \frac{9}{169}$

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An urn has three red marbles and eight blue marbles in it. Draw two marbles, one at a time, this time without replacement, from the urn. "Without replacement" means that you do not put the first ball back before you select the second marble. Following is a tree diagram for this situation. The branches are labeled with probabilities instead of frequencies. The numbers at the ends of the branches are calculated by multiplying the numbers on the two corresponding branches, for example, $(\frac{3}{11}) (\frac{2}{10}) = \frac{6}{110}$ .

This is a tree diagram with branches showing probabilities of each draw. The first branch shows 2 lines: B 8/11 and R 3/11. The second branch has a set of 2 lines for each first branch line. Below B 8/11 are B 7/10 and R 3/10. Below R 3/11 are B 8/10 and R 2/10. Multiply along each line to find BB 56/110, BR 24/110, RB 24/110, and RR 6/110. — Total = $\frac{56 + 24 + 24 + 6}{110} = \frac{110}{110} = 1$

Note

If you draw a red on the first draw from the three red possibilities, there are two red marbles left to draw on the second draw. You do not put back or replace the first marble after you have drawn it. You draw without replacement , so that on the second draw there are ten marbles left in the urn.

Calculate the following probabilities using the tree diagram.

a. P ( RR ) = ________

a. P ( RR ) = $(\frac{3}{11}) (\frac{2}{10}) = \frac{6}{110}$

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b. Fill in the blanks:

P ( RB OR BR ) = $(\frac{3}{11}) (\frac{8}{10}) + (___)(___) = \frac{48}{110}$

b. P ( RB OR BR ) = $(\frac{3}{11}) (\frac{8}{10})$ + $(\frac{8}{11}) (\frac{3}{10})$ = $\frac{48}{110}$

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c. P ( R on 2nd| B on 1st) =

c. P ( R on 2nd| B on 1st) = $\frac{3}{10}$

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d. Fill in the blanks.

P ( R on 1st AND B on 2nd) = P ( RB ) = (___)(___) = $\frac{24}{100}$

d. P ( R on 1st AND B on 2nd) = P ( RB ) = $(\frac{3}{11}) (\frac{8}{10})$ = $\frac{24}{100}$

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e. Find P ( BB ).

e. P ( BB ) = $(\frac{8}{11}) (\frac{7}{10})$

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f. Find P ( B on 2nd| R on 1st).

f. Using the tree diagram, P ( B on 2nd| R on 1st) = P ( R | B ) = $\frac{8}{10}$ .

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If we are using probabilities, we can label the tree in the following general way.

This is a tree diagram for a two-step experiment. The first branch shows first outcome: P(B) and P(R). The second branch has a set of 2 lines for each line of the first branch: the probability of B given B = P(BB), the probability of R given B = P(RB), the probability of B given R = P(BR), and the probability of R given R = P(RR).

P ( R | R ) here means P ( R on 2nd| R on 1st)
P ( B | R ) here means P ( B on 2nd| R on 1st)
P ( R | B ) here means P ( R on 2nd| B on 1st)
P ( B | B ) here means P ( B on 2nd| B on 1st)

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Questions & Answers

how did you get 1640

Noor Reply

If auger is pair are the roots of equation x2+5x-3=0

Peter Reply

Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.

MATTHEW Reply

420

Sharon

from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph

George

Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28

Salma

Devon is 32 32 years older than his son, Milan. The sum of both their ages is 54 54. Using the variables d d and m m to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.

Aaron Reply

find product (-6m+6) ( 3m²+4m-3)

SIMRAN Reply

-42m²+60m-18

Salma

what is the solution

bill

how did you arrive at this answer?

bill

-24m+3+3mÁ^2

Susan

i really want to learn

Amira

I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please

Amira

Hw did u arrive to this answer.

Aphelele

Bajemah

-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18

Salma

complete the table of valuesfor each given equatio then graph. 1.x+2y=3

Jovelyn Reply

x=3-2y

Salma

y=x+3/2

Salma

Enock

given that (7x-5):(2+4x)=8:7find the value of x

Nandala

3x-12y=18

Kelvin

please why isn't that the 0is in ten thousand place

Grace Reply

please why is it that the 0is in the place of ten thousand

Grace

Send the example to me here and let me see

Stephen

A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg

Marry Reply

how far

Abubakar

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Enock

state in which quadrant or on which axis each of the following angles given measure. in standard position would lie 89°

Abegail Reply

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BenJay

Method

I am eliacin, I need your help in maths

Rood

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Sir

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Amoon

however, may I ask you some questions about Algarba?

Amoon

Enock

what the last part of the problem mean?

Roger

The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.

cameron Reply

Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.

mahnoor Reply

I'm guessing, but it's somewhere around $4335.00 I think

Lewis

12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00

Munster

difference between rational and irrational numbers

Arundhati Reply

When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?

Jakoiya Reply

how to reduced echelon form

Solomon Reply

Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?

Zack Reply

d=r×t the equation would be 8/r+24/r+4=3 worked out

Sheirtina

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Source: OpenStax, Introductory statistics. OpenStax CNX. May 06, 2016 Download for free at http://legacy.cnx.org/content/col11562/1.18

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