Probability & 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 For example – a fair coin is spun twice 1st 2nd H HH T HT H TH T TT H T Possible Outcomes Attach probabilities 1st ½ ½ 2nd ½ H H HH P(H,H)=½x½=¼ T HT P(H,T)=½x½=¼ ½ H TH P(T,H)=½x½=¼ ½ T TT P(T,T)=½x½=¼ ½ T INDEPENDENT EVENTS – 1st spin has no effect on the 2nd spin Calculate probabilities 1st ½ ½ 2nd ½ H H HH P(H,H)=½x½=¼ * T HT P(H,T)=½x½=¼ ½ H TH P(T,H)=½x½=¼ * * ½ T TT P(T,T)=½x½=¼ ½ T Probability of at least one Head? 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 B G R RR B RB G R RG BR B BB G R BG GR B GB G GG INDEPENDENT EVENTS Probabilities 1st 2nd 0.3 0.2 0.3 R 0.5 0.3 0.2 0.2 B 0.5 0.5 0.3 0.2 G 0.5 R RR P(RR) = 0.3x0.3 = 0.09 B RB P(RB) = 0.3x0.2 = 0.06 G R RG BR P(RG) = 0.3x0.5 = 0.15 P(BR) = 0.2x0.3 = 0.06 B BB P(BB) = 0.2x0.2 = 0.04 G R BG GR P(BG) = 0.2x0.5 = 0.10 P(GR) = 0.5x0.3 = 0.15 B GB P(GB) = 0.5x0.2 = 0.10 G GG P(GG) = 0.5x0.5 = 0.25 All ADD UP to 1.0 Choose a meal Main course Salad 0.2 Egg & Chips 0.5 Pizza 0.3 IC Pudding Ice Cream 0.45 Apple Pie 0.55 P(S,IC) = 0.2 x 0.45 = 0.09 AP P(S,AP) = 0.2 x 0.55 = 0.110 0.45 IC P(E,IC) = 0.5 x 0.45 = 0.225 0.55 AP P(E,AP) = 0.5 x 0.55 = 0.275 IC P(P,IC) = 0.3 x 0.45 = 0.135 AP P(P,AP) = 0.3 x 0.55 = 0.165 0.45 S 0.55 0.2 0.5 E 0.3 P 0.45 0.55