Tree diagrams

What are Tree Diagrams A way of showing the possibilities of two or more events Simple diagram we use to calculate the probabilities of two or more events

A fair coin is spun twice 1st 2nd H HH Possible Outcomes H T HT H TH T T TT

Attach probabilities 1st 2nd H HH P(H,H)=½x½=¼ ½ ½ H ½ T HT P(H,T)=½x½=¼ ½ H TH P(T,H)=½x½=¼ ½ T ½ T TT P(T,T)=½x½=¼ INDEPENDENT EVENTS – 1st spin has no effect on the 2nd spin

* * * Calculate probabilities 1st 2nd H HH P(H,H)=½x½=¼ ½ ½ H ½ T HT P(H,T)=½x½=¼ * ½ H TH P(T,H)=½x½=¼ ½ T ½ T TT P(T,T)=½x½=¼ Probability of at least one Head?

1st 2nd R RR B RB R G RG R BR B B BB G BG R GR G GB B G GG For example – 10 coloured beads in a bag – 3 Red, 2 Blue, 5 Green. One taken, colour noted, returned to bag, then a second taken. 1st 2nd R RR B RB R G RG R BR INDEPENDENT EVENTS B B BB G BG R GR G GB B G GG

Probabilities 1st 2nd R RR B RB R G RG R BR B B BB G BG R GR G GB B G P(RR) = 0.3x0.3 = 0.09 0.3 0.2 B RB P(RB) = 0.3x0.2 = 0.06 R 0.3 0.5 G RG P(RG) = 0.3x0.5 = 0.15 R BR P(BR) = 0.2x0.3 = 0.06 0.3 0.2 0.2 B B BB P(BB) = 0.2x0.2 = 0.04 0.5 G BG P(BG) = 0.2x0.5 = 0.10 R GR 0.3 P(GR) = 0.5x0.3 = 0.15 0.5 G 0.2 GB B P(GB) = 0.5x0.2 = 0.10 G GG 0.5 P(GG) = 0.5x0.5 = 0.25 All ADD UP to 1.0

Dependent Event What happens the during the second event depends upon what happened before. In other words, the result of the second event will change because of what happened first. The probability of two dependent events, A and B, is equal to the probability of event A times the probability of event B. However, the probability of event B now depends on event A. Slide 8

Dependent Event Example: There are 6 black pens and 8 blue pens in a jar. If you take a pen without looking and then take another pen without replacing the first, what is the probability that you will get 2 black pens? P(black first) = P(black second) = (There are 13 pens left and 5 are black) THEREFORE……………………………………………… P(black, black) = Slide 9

Dependent Events Find the probability 1 26 25 650 P(Q, Q) All the letters of the alphabet are in the bag 1 time Do not replace the letter 1 26 25 650 x = Slide 10

Are these dependent or independent events? TEST YOURSELF Are these dependent or independent events? Tossing two dice and getting a 6 on both of them. 2. You have a bag of marbles: 3 blue, 5 white, and 12 red. You choose one marble out of the bag, look at it then put it back. Then you choose another marble. 3. You have a basket of socks. You need to find the probability of pulling out a black sock and its matching black sock without putting the first sock back. 4. You pick the letter Q from a bag containing all the letters of the alphabet. You do not put the Q back in the bag before you pick another tile. Slide 11

7 Red 3 Blue. Pick 2, without replacement 7 Red 3 Blue. Pick 2, without replacement. a) p(R,R) b) p(B,B) c) p(One of each) OUTCOMES P(Outcome) 2nd event 1st event 6/9 R,R P(R,R)=42/90 7/10 3/10 3/9 R,B P(R,B)=21/90 7/9 B,R P(B,R)=21/90 2/9 B,B P(B,B)=6/90 Total P(all outcomes) = 1

Probability Trees Example 1 A bag contains 6 red beads and 4 blues. 2 beads are picked at random without replacement. (i) Draw a probability tree diagram to show this information (ii) Calculate the probability of selecting both red beads (iii) Calculate the probability of picking one of each colour.

R R B R B B Probability Trees Example 1 1st Pick 2nd Pick A bag contains 6 red beads and 4 blues. 2 beads are picked at random without replacement. (i) Draw a probability tree diagram to show this information (ii) Calculate the probability of selecting both red beads (iii) Calculate the probability of picking one of each colour. 1st Pick 2nd Pick R R B R B B

R R B R B B Probability Trees Example 1 1st Pick 2nd Pick ? ? ? ? ? ? A bag contains 6 red beads and 4 blues. 2 beads are picked at random without replacement. (i) Draw a probability tree diagram to show this information (ii) Calculate the probability of selecting both red beads (iii) Calculate the probability of picking one of each colour. 1st Pick 2nd Pick R ? R ? B ? R ? ? B ? B To Part (ii)

R R B R B B Probability Trees Example 1 1st Pick 2nd Pick A bag contains 6 red beads and 4 blues. 2 beads are picked at random without replacement. (i) Draw a probability tree diagram to show this information (ii) Calculate the probability of selecting both red beads (iii) Calculate the probability of picking one of each colour. 1st Pick 2nd Pick R R B R B B

R R B R B B Probability Trees Example 1 1st Pick 2nd Pick A bag contains 6 red beads and 4 blues. 2 beads are picked at random without replacement. (i) Draw a probability tree diagram to show this information (ii) Calculate the probability of selecting both red beads (iii) Calculate the probability of picking one of each colour. 1st Pick 2nd Pick R R B R B B

Probability Trees Question 1 A bag contains 7 red beads and 3 blues. 2 beads are picked at random without replacement. (i) Draw a probability tree diagram to show this information (ii) Calculate the probability of selecting both red beads (iii) Calculate the probability of picking one of each colour.

R R B R B B Probability Trees Question 1 1st Pick 2nd Pick ? ? ? ? ? ? A bag contains 7 red beads and 3 blues. 2 beads are picked at random without replacement. (i) Draw a probability tree diagram to show this information (ii) Calculate the probability of selecting both red beads (iii) Calculate the probability of picking one of each colour. 1st Pick 2nd Pick R ? R ? B ? ? R ? B ? B To Part (ii)

R R B R B B Probability Trees Question 1 1st Pick 2nd Pick A bag contains 7 red beads and 3 blues. 2 beads are picked at random without replacement. (i) Draw a probability tree diagram to show this information (ii) Calculate the probability of selecting both red beads (iii) Calculate the probability of picking one of each colour. 1st Pick 2nd Pick R R B R B B To Part (iii)

R R B R B B Probability Trees Question 1 1st Pick 2nd Pick A bag contains 7 red beads and 3 blues. 2 beads are picked at random without replacement. (i) Draw a probability tree diagram to show this information (ii) Calculate the probability of selecting both red beads (iii) Calculate the probability of picking one of each colour. 1st Pick 2nd Pick R R B R B B

Probability Trees Question 2 A bag contains 4 yellow beads and 3 blues. 2 beads are picked at random without replacement. (i) Draw a probability tree diagram to show this information (ii) Calculate the probability that both beads selected will be blue (iii) Calculate the probability of picking one of each colour.

Y Y B Y B B Probability Trees Solution 2 1st Game 2nd Game A bag contains 4 yellow beads and 3 blues. 2 beads are picked at random without replacement. (i) Draw a probability tree diagram to show this information (ii) Calculate the probability that both beads selected will be blue (iii) Calculate the probability of picking one of each colour. 1st Game 2nd Game Y Y B Y B B

Probability (Tree Diagrams) Question 5 Lucy has a box of 30 chocolates. 18 are milk chocolate and the rest are dark chocolate. She takes a chocolate at random from the box and eats it. She then chooses a second. (a) Draw a tree diagram to show all the possible outcomes. (b) Calculate the probability that Lucy chooses: (i) 2 milk chocolates. (ii) A dark chocolate followed by a milk chocolate. Probability (Tree Diagrams) Dependent Events Milk Dark First Pick Second Pick Q5 Chocolates

S S R S R R Probability Trees Question 3 The probability that Stuart wins a game of darts against Rose is 0.7. They play two games. (i) Copy & complete the probability tree diagram shown below (ii) Calculate the probability Rose winning both games (iii) Calculate the probability of the final result being a draw. 1st Game 2nd Game S S R S R R

S S R S R R Probability Trees Solutions 3 The probability that Stuart wins a game of darts against Rose is 0.7. They play two games. (i) Copy & complete the probability tree diagram shown below (ii) Calculate the probability Rose winning both games (iii) Calculate the probability of the final result being a draw. 1st Game 2nd Game S S R S R R

Independent Practice Solve #1, 3, 4 on pages 370 -371 (Exercise 8P) For review (IB Test and non IB registered students) – Use exam style questions on pages 372 – 376.