GCSE: Tree Diagrams and Frequency Trees Skipton Girls’ High School

!Mutually Exclusive Events If A and B are mutually exclusive events, they can’t happen at the same time. Then: P(A or B) = P(A) + P(B) ! Independent Events If A and B are independent events, then the outcome of one doesn’t affect the other. Then: P(A and B) = P(A) × P(B)

Tree Diagrams Unconditional Question: There are 3 blues and 5 reds on a spinner – There are 2 spins. Spin 2 P(bb)  3/8 x 3/8 = 9/64 Spin 1 P(blue) = 3/8 P(blue) = 3/8 P(br)  3/8 x 5/8 = 15/64 P(red) = 5/8 P(rb)  5/8 x 3/8 = 15/64 P(blue) = 3/8 P(red) = 5/8 P(red) = 5/8 P(rr)  5/8 x 5/8 = 25/64 Check: 25 + 15 + 15 + 9 = 64

R R B R B B Tree Diagrams Conditional 4 ? 6 5 ? 7 2 ? 6 5 ? 6 2 ? 7 1 Question: Given there’s 5 red balls and 2 blue balls. What’s the probability that after two picks we have a red ball and a blue ball? Key Point: Note that probabilities go on the lines, and events at the end. After first pick, there’s less balls to choose from, so probabilities change. 4 6 ? R ? 5 7 R B ? 2 6 5 6 R ? 2 7 ? B 1 6 B ?

Tree Diagrams Question: Give there’s 5 red balls and 2 blue balls. What’s the probability that after two picks we have a red ball and a blue ball? We multiply across the matching branches, then add these values. 4 6 R 5 7 R 5 21 ? B 2 6 5 21 5 6 R ? 2 7 B 1 6 B 10 21 ? P(red and blue) =

Conditional and Unconditional Tree Diagrams ...with replacement: The item is returned before another is chosen. The probability of each event on each trial is fixed. ...without replacement: The item is not returned. Total balls decreases by 1 each time. Number of items of this type decreases by 1. Note that if the question doesn’t specify which, e.g. “You pick two balls from a bag”, then PRESUME WITHOUT REPLACEMENT.

Example (on your sheet) ? 3 8 ? 3 8 ? 5 8 ? 5 8 × 5 8 = 25 64 ? 3 8 ? 5 8 3 8 × 5 8 + 5 8 × 3 8 = 15 32 ?

Question 1 ? 1 5 × 1 5 = 1 25 1 5 × 4 5 + 4 5 × 1 5 = 8 25 ? 1− 8 25 = 17 25 ?

Question 2 ? 0.9 ? 0.9 0.1 0.9 ? 0.1 0.1 ? 0.9 2 =0.81 ? 2×0.1×0.9=0.18

Question 3 ? 4 13 5 14 ? 9 13 5 13 ? 9 14 8 13 9 14 × 8 13 = 36 91 ? two consonants? 5 14 × 9 13 + 9 14 × 5 13 = 45 91 ?

Question 4 2 9 ? ? 3 10 7 9 3 9 ? 7 10 6 9 3 10 × 7 9 + 7 10 × 3 9 = 7 30 ? 1− 7 30 = 23 30 ?

Frequency Trees 30 students were asked if they liked coffee. 20 of the students were girls. 6 boys liked coffee. 12 girls did not like coffee. Like coffee 8 20 girls 12 Don’t like coffee 30 6 Like coffee boys 10 4 Don’t like coffee

56 students were asked if they watched tennis yesterday. 20 of the students are boys. 13 boys did not watch tennis. 17 girls watched tennis. Did watch tennis 17 36 girls 19 Didn’t watch tennis 56 7 Did watch tennis boys 20 13 Didn’t watch tennis

85 students were asked what they did for lunch, 47 of these were female. 5 boys had a packed lunch. 21 girls had a school dinner. 13 girls had ‘other’. Altogether 40 pupils had a school dinner. 13 Packed lunch 47 School dinner 21 female other 13 85 5 Packed lunch male 38 School dinner 19 14 other

18 47 14 15 80 8 33 19 6 80 Year 12 students each study one Science. 47 of these students are females. 18 girls study Biology, and 19 boys study physics. Altogether 33 study physics and 21 pupils study chemistry. 18 Biology 47 Physics 14 female chemistry 15 80 8 Biology male 33 Physics 19 6 Chemistry