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1 PLC Papers Created For: Year 11 Topic Practice Paper: Tree Diagrams (Conditional Probability)

2 Conditional Probability 1 Grade 7 Objective: Calculate and interpret conditional probabilities, using expected frequencies with twoway tables, tree diagrams and Venn diagrams Question 1. A bag of sweets contains 9 mints, 6 toffees and 5 sherbert lemons. Helen takes 3 sweets at random from the bag. Work out the probability that, of the three sweets Helen takes, exactly two will be the same flavour. (4) (Total 4 marks)

3 Question 2. A box contains 3 new batteries, 5 partly used batteries and 4 dead batteries. Kelly takes two batteries at random. Work out the probablitity that she picks two dead batteries. (3) (Total 3 marks)

4 Question 3 Caleb either walks to school or travels by bus. The probability that he walks to school is If he walks to school, the probability that he will be late is 0.3. If he travels to school by bus, the probability that he will be late is 0.1. Work out the probability that he will not be late. (3) (Total 3 marks) Total /10

5 Conditional Probability 2 Grade 7 Objective: Calculate and interpret conditional probabilities, using expected frequencies with twoway tables, tree diagrams and Venn diagrams Question 1. In a group of students, 45% are girls. 65% of these prefer to play tennis rather than badminton. 10% of the boys prefer to play badminton rather than tennis. One student is chosen at random. Find the probability that this is a boy who prefers to play tennis. (2) (Total 2 marks)

6 Question 2. Laura has 9 tins of soup in her cupboard, but all the labels are missing. She knows that there are 5 tins of tomato soup and 4 tins of vegetable soup. She opens three tins at random. Work out the probability that she opens more tins of vegetable soup than tomato soup. (4)

7 Question 3 (Total 4 marks) Steve has to catch a flight. The probability of dry weather (D), rain (R) or snow (S) are: P(D) = 0.6, P(R) = 0.35, P(S) = If it is dry the probability that Steve will arrive to the airport on time is 0.9. If it rains the probability that he will arrive to the airport on time is 0.6. If it snows the probability that he will arrive to the airport on time is Is he more likely to arrive on time to the airport or be late? (4) (Total 4 marks)

8 Total /10

9 Conditional Probability 3 Grade 7 Objective: Calculate and interpret conditional probabilities, using expected frequencies with twoway tables, tree diagrams and Venn diagrams Question 1. The two way table shows the number of deaths and serious injuries caused by road traffic accidents in Great Britain in Work out an estimate for the probability: (a) that the accident is serious.... (b) that the accident is fatal given that the speed limit is 30 mph. (1)... (c) that the accident happens at 20 mph given that the accident is serious. (2)... (2) (Total 5 marks)

10 Question 2. Bag A contains 5 red counters and 4 green counters. Bag B contains 3 red counters and 6 green counters. Move 1 A counter is taken from bag A and placed into bag B. Move 2 A counter is taken from bag B and placed into bag A. Work out the probability that bag A has more red counters than green counters after these two moves. (5) (Total 5 marks)

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12 Conditional Probability 4 Grade 7 Objective: Calculate and interpret conditional probabilities, using expected frequencies with twoway tables, tree diagrams and Venn diagrams Question 1. The Venn diagram shows the ice-cream flavours chosen by a group of 44 children at a party. The choices are strawberry (S), choc-chip (C) and toffee (T). A child is picked at random. Work out : (a) P(S)... (b) P(T U C C) (1)... (c) P(C S U T) (2)... (2)

13 Question 2. Max has an empty box. He puts some red counters and some blue counters into the box. The ratio of the number of red counters to the number of blue counters is 1 : 3. Julie takes at random 2 counters from the box. The probability that she takes 2 red counters is How many red counters did Max put in the box? (Total 5 marks) (5) (Total 5 marks) Total /10

14 PLC Papers Created For: Year 11 Topic Practice Paper: Tree Diagrams (Conditional Probability)

15 Conditional Probability 1 Grade 7 SOLUTIONS Objective: Calculate and interpret conditional probabilities, using expected frequencies with twoway tables, tree diagrams and Venn diagrams Question 1. A bag of sweets contains 9 mints, 6 toffees and 5 sherbert lemons. Helen takes 3 sweets at random from the bag. Work out the probability that, of the three sweets Helen takes, exactly two will be the same flavour. MMT MMS TTM TTS SSM SST 9 x 8 x 6 x 3 = x 8 x 5 x 3 = x 5 x 9 x 3 = x 5 x 5 x 3 = x 4 x 9 x 3 = x 4 19 x 6 18 x 3 = 1 19 Correct outcomes chosen Multiplying each probability by 3 Adding their probabilities Correct solution A1 P(two the same flavour) = Or 9 MMM x 8 x 7 = TTT x 5 x 4 = SSS x 4 x 3 = MTS x 6 x 5 x 6 = P(two the same flavour) = = Correct outcomes chosen Multiplying probability of MTS by 6 Subtracting their answer from 1 Correct solution A1 (4) (Total 4 marks)

16 Question 2. A box contains 3 new batteries, 5 partly used batteries and 4 dead batteries. Kelly takes two batteries at random. Work out the probablitity that she picks two different types of batteries. NP ND PN PD DN DP 3 12 x 5 11 = x 4 11 = x 3 11 = x 4 11 = x 3 11 = x 5 11 = 5 33 Multiplying each probability Adding their probabilities Correct solution A1 P(two different types) = Or 3 NN x 2 = PP x 4 = DD x 3 = P(two the same flavour) = 1 19 = Multiplying probability of MTS by 6 Subtracting their answer from 1 Correct solution A1 (3) (Total 3 marks)

17 Question 3 Caleb either walks to school or travels by bus. The probability that he walks to school is If he walks to school, the probability that he will be late is 0.3. If he travels to school by bus, the probability that he will be late is 0.1. Work out the probability that he will not be late x 0.7 = or 0.25 x 0.9 = = 0.75 A1 (3) (Total 3 marks) Total /10

18 Conditional Probability 2 Grade 7 SOLUTIONS Objective: Calculate and interpret conditional probabilities, using expected frequencies with twoway tables, tree diagrams and Venn diagrams Question 1. In a group of students, 45% are girls. 65% of these prefer to play tennis rather than badminton. 10% of the boys prefer to play badminton rather than tennis. One student is chosen at random. Find the probability that this is a boy who prefers to play tennis x 0.9 Multiply probabilities together Correct solution A1 oe (2) (Total 2 marks)

19 Question 2. Laura has 9 tins of soup in her cupboard, but all the labels are missing. She knows that there are 5 tins of tomato soup and 4 tins of vegetable soup. She opens three tins at random. Work out the probability that she opens more tins of vegetable soup than tomato soup. TVV VTV VVT 5 9 x 4 8 x 3 7 = x 5 8 x 3 7 = x 3 8 x 5 7 = 5 42 VVV 4 9 x 3 8 x 2 7 = 1 21 Correct outcomes chosen Multiplying each probability Adding their probabilities Correct solution A1 P(more vegetable) = (4) (Total 4 marks)

20 Question 3. Steve has to catch a flight. The probability of dry weather (D), rain (R) or snow (S) are: P(D) = 0.6, P(R) = 0.35, P(S) = If it is dry the probability that Steve will arrive to the airport on time is 0.9. If it rains the probability that he will arrive to the airport on time is 0.6. If it snows the probability that he will arrive to the airport on time is Is he more likely to arrive on time to the airport or be late? P(on time) = (0.6 x 0.9) + (0.35 x 0.6) + (0.05 x 0.15) P(late) = (0.6 x 0.1) + (0.35 x 0.4) + (0.05 x 0.85) P(on time) = 0.76 and P(late) = 0.24 A1 Steve is more likely to be on time. C1 (4) (Total 4 marks) Total /10

21 Conditional Probability 3 Grade 7 SOLUTIONS Objective: Calculate and interpret conditional probabilities, using expected frequencies with twoway tables, tree diagrams and Venn diagrams Question 1. The two way table shows the number of deaths and serious injuries caused by road traffic accidents in Great Britain in Work out an estimate for the probability: (a) that the accident is serious or 0.95 A1... (1) (b) that the accident is fatal given that the speed limit is 30 mph = or 0.04 M2 (Correct working must be seen) Allow for = or 0.04 (c) that the accident happens at 20 mph given that the accident is serious. 420 = or 0.03 M2 (Correct working must be seen) (2) Allow for = or (2)

22 (Total 5 marks) Question 2. Bag A contains 5 red counters and 4 green counters. Bag B contains 3 red counters and 6 green counters. Move 1 A counter is taken from bag A and placed into bag B. Move 2 A counter is taken from bag B and placed into bag A. Work out the probability that bag A has more red counters than green counters after these two moves. Identify probabilities as 9, 10, 9 RRR RRG GRR GRG GGR GGG 5 x 4 x 5 = x 4 10 x 4 9 = x 3 10 x 6 9 = x 3 x 3 = x 7 x 5 = x 7 x 4 = Correct outcomes chosen Multiplying each probability Adding their probabilities Correct solution A1 P(more red) = 2 3 (5) (Total 5 marks) Total /10

23 Conditional Probability 4 Grade 7 SOLUTIONS Objective: Calculate and interpret conditional probabilities, using expected frequencies with twoway tables, tree diagrams and Venn diagrams Question 1. The Venn diagram shows the ice-cream flavours chosen by a group of 44 children at a party. The choices are strawberry (S), choc-chip (C) and toffee (T). A child is picked at random. Work out : (a) P(S) = 9 22 (b) P(T U C C) 7 23 A1... M2 (1) (Allow for 7 or 23 )... (2) (c) P(C S U T) (Allow for 14 or 27 ) M2... (2)

24 Question 2. Max has an empty box. He puts some red counters and some blue counters into the box. The ratio of the number of red counters to the number of blue counters is 1 : 3. Julie takes at random 2 counters from the box. The probability that she takes 2 red counters is How many red counters did Max put in the box? (Total 5 marks) For process to start to solve. E.g. use of x and 3x To form fractions for each probability. E.g. 4 Process to form equation e.g. 4 x = and, Process to eliminate fractions and reduce equation to linear form E.g. 316x 316 = 304x A1 (5) (Total 5 marks) Total /10