# Worksheets for GCSE Mathematics. Probability. mr-mathematics.com Maths Resources for Teachers. Handling Data

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1 Worksheets for GCSE Mathematics Probability mr-mathematics.com Maths Resources for Teachers Handling Data

2 Probability Worksheets Contents Differentiated Independent Learning Worksheets Probability Scales Page 30 Writing Probabilities Page 30 Expected Frequencies Page 50 Sample Space Diagrams Page 60 Mutually Exclusive Events Page 70 Probability Trees Page 80 Venn Diagrams Page 90 Conditional Events Page 10 Union and Intersection of Sets Page 11 Solutions Probability Scales Writing Probabilities Expected Frequencies Sample Space Diagrams Mutually Exclusive Events Probability Trees Venn Diagrams Conditional Events Page 12 Page 13 Page 14 Page 15 Page 16 Page 17 Page 18 Page 19 Union and Intersection of Sets Page 20

3 Probability Scale Q1. Use these keywords to describe the likelihood of the following events. It will be Tuesday the day after Thursday. b) A fair coin will land on heads. c) The sun will come up today. d) It will rain tomorrow. e) You will arrive on time to school tomorrow. f) You will celebrate your 60 th birthday tomorrow. g) The next car you see will be a Ferrari. Q2. LE Suzie reaches into the box to pick a ball at random. Place these events along the probability scale to describe their likelihood. Picking a green ball. b) Picking a coloured ball. c) Picking a red ball. d) Picking a red or black ball. e) Picking an orange coloured ball. Q3. Write in six different events that can be placed as shown along the probability scale below. 3

4 Writing Probabilities Q1. A ball is taken at random from this box. Write the following probabilities; leave your answer as a simplified fraction. P (Red) b) P (Blue) c) P (Purple) d) P (Red or purple) e) P (Black) f) P (Not Black) Q2. The arrow is spun around the spinner so it lands randomly on a number. Write the following probabilities; leave your answer as a simplified fraction. P (5 or 6) b) P (Odd number) LE c) P (Less than 6) d) P (Greater than 7) e) P (Factor of 12) f) P (Multiple of 3) Q3. Here are two spinners. Which spinner has P (White) =? b) Which spinner has P (Blue) =? c) Which spinner has P (1) =? d) Which spinner has P (2) =? Q4. An ordinary six sided dice is rolled. Work out the probability of getting a 4 b) an even number c) a 8 d) a number greater than 2 e) a number less than 7 f) zero. Q5. A bag contains red and blue counters only. The probability of pick a red is. What is the probability of picking a blue? 4

5 Q1. A fair six sided dice is thrown. What is the probability of the throwing a Expected Frequency i) Three ii) One iii) Prime Number iv) Less than six b) Land on a 4 number after 300 throws? c) Land on an even number after 120 throws? d) Land on a prime number after 120 throws? e) Land on a number less than six after 240 throws? Q2. A spinner, with 12 equal sections, is spun 240 times. How often would you expect to spin: a shaded section b) an even number c) number 3 d) square number e) a prime number and shaded section LE Q3. This pentagonal spinner is spun 100 times with the results recorded. Here are the results. What is the probability of the spinner landing on the colour i) Red ii) Green iii) Blue iv) White? b) How many times would you expect the spinner to land on each colour after 500 spins? c) How many times would you expect the spinner to land on each colour after 1000 spins? Q4 A coin is biased so that the probability of tossing a head is 0.7. How many heads would you expect when the coin is tossed 100 times? b) How many tails would you expect when the coin is tossed 300 times? Q5. Players, at a village fayre, pay 1 to spin the pointed on the board shown. Players win the amount shown by the pointer. The game is played 600 times. Work out the expected profit or loss on the game. 5

6 Q1. Sample Space Diagrams 20p and 50p coins are tossed at the same time. List all the possible outcomes. b) Calculate the probability of tossing two heads. c) Calculate the probability of tossing a head or tails in any order. Q2. The spinner shown has three equal sections on the inside and six on the outside. List all the possible outcomes. b) Calculate the probability of spinning a i) number 2 ii) C iii) 3 and B iv) 5 and A v) C and 6 vi) Not B and 3 LE Q3. A fair coin is tossed and a fair dice is rolled. Q4. Complete the sample space diagram to show all the possible outcomes. What is the probability of obtaining: a head and six b) a tail and odd number c) a head and greater than 2 d) a tail and less than 3 e) a prime number f) a head and 5 or tail and 1 The diagram shows a fair spinner dividing into three equal sections. The spinner is spun twice and the numbers are added together. Complete a sample space diagram to show all the possible scores. b) Calculate the probability of getting a score of: i) 2 ii) 5 iii) square number iv) odd number v) Less than 10 vi) Greater than 6 Q5. The diagram shows two sets of card Hearts and Clubs. One card is taken at random from each set. List all the possible outcomes. b) What is the probability of 6 i) Ace and 8 ii) Hearts and 7 iii) Clubs and 2 iv) Hearts and Clubs

7 Mutually Exclusive Events Q1. The scale shows the probability that a fair six sided dice will land on the number 5. Calculate the probability of the dice not landing on a five. Q2. The probability that Adam wins a raffle is 15%. Calculate the probability that Adam does win the raffle. Q3. A bad contains red, black and blue balls. The probability of picking a red ball is 0.4. The probability of picking a black ball is Calculate the probability of picking a black ball. Q4. A bag contains blue, red, green orange and black marbles. The table shows the probabilities of picking each marble at random. LE How can you tell there is a mistake in the table? b) The probability of picking a red marble is wrong. What should it be? c) What is the probability of picking a marble that is black, orange or red? d) What is the probability of picking a marble that is not blue? Q5. A spinner is shown with the probabilities of landing on each colour recorded. What is the probability of the spinner landing on one of the following? Not Pink b) Pink c) Not Green d) Not Red Q6. A bag contains only red, blue, green, black and purple counters. The probabilities for picking blue, black and purple counters are recorded. The probabilities of picking a green and red counter are equal. Calculate the probability of picking a red and green counter. b) The spinner is spun 300 times. How many reds would you expect to record? 7

8 Q1. b) c) d) Q2. b) c) d) e) Q3. b) c) d) Probability Trees A bag contains only red and blue counters. There are 4 red and 1 blue. Paul takes a counter from the bag at random and replaces it. Complete the tree diagram to show the possible outcomes. Calculate the probability of: taking out two red counters. taking out two different colours. not taking a red counter. The train network has stated that 85% of trains arrive on time. Gemma has to take two trains on her way to work. Complete the tree diagram. LE What is the probability that: the first train is on time but the second one is late. at least one train is late. both trains are late. neither train is late. Along Joanna s drive to work she crosses two traffic lights that are independent of each other. The probability she is stuck at the first light is 0.6. The probability of being stopped at the second light is 0.5. Calculate the probability that on a given day: she stops at both sets of lights. Joanna stops at the first set but not the other. Joanna does not have to stop. she stops for at least one set. Q4. A couple has three children. It is equally likely that each child is a boy or girl. Draw a probability tree to show the possible genders of each baby. Use the probability tree to calculate: all three are girls b) they have one boy and two girls 8 c) they have at least two boys. d) they have no girls. e) they have at least one girl. e) they have at least one boy and one girl.

9 Q1. Venn Diagrams The Venn diagrams show the subjects taken by a group of 60 college students. Work out the value of x in each case. b) c) Q2. In form 10J 15 students are studying Maths, 12 students are studying French, 6 are studying both and 3 students study neither. Complete a Venn Diagram to represent this data. b) How many students are there in the form? A student is chosen at random. c) Calculate the probability that the students chooses Maths or French but not both. LE Q3. Of the households in the United Kingdom 35% have a L.E.D. TV and 60% have a Blu-ray player. There are 24% that have both. Complete a Venn Diagram to show this information. b) Calculate the probability that a household chosen at random has either a L.E.D. TV or a Bluray player but not both. c) Calculate the probability that a household chosen at random has neither a L.E.D. TV nor a Bluray player. Q4. A survey of 100 people at a local gym were asked how they spend their time there. 34 people used the free weights. 28 people used cardiovascular equipment 38 people went to the classes. There were 9 who used the weights and went to classes, 5 who used the cardiovascular equipment and went to classes and 10 who used the free weights and cardiovascular equipment. 4 people made use of all three facilities. Represent these data on a Venn diagram. A person was selected at random from this group. Find the probability that this person b) i) used the free weights but did not use the cardiovascular equipment. ii) went to the classes but did not use free weights. ii) only used two of the three facilities. 9

10 Conditional Probability Q1. A bag contains three white balls and two black. A ball is taken out at random and not replaced. Another is then taken out. Draw a tree diagram to calculate the probability that: both balls will be white. b) one ball of each colour is removed. Q2. Of the 15 M&Ms left in a bag, 8 are red and the rest are blue. Dominic chooses two of these random, one after the other. Draw a tree diagram to calculate the probability that Dominic chooses: both red. b) one of each colour. c) both the same colour d) at least one blue. Q3. On Kelly s way to work she passes two sets of traffic lights. The probability that the first is green is. If the first is green, the probability that the second is green is. If the first is red the probability that the second is green is. LE What is the probability that: both are red b) none are red c) only one is green d) at least one is green. Q4. The probability of passing a test at the first attempt is Those who fail are allowed a re-sit. The probability of passing the re-sit is 0.8. What is the probability of failing at the second attempt? Q5. 65% of people over the age of 60 have the flu vaccination. Of those who do not have the vaccination 75% get the flu. Calculate the probability that a person over the age of 60 will get the flu. Q6. Pauline has twenty biscuits in a tin. 6 are ginger, 12 are plain and the rest are chocolate. Pauline takes two biscuits out of the tin. Draw a tree diagram to work out the probability the two were not the same type. Q7. There are three different types of sandwiches in a shop. There are 5 salads, 4 cheese and 2 ham. Jodie takes two sandwiches at random. Calculate the probability that she takes two sandwiches of the same type. Q8. There are 8 coins in a box. Five and 1 coins and three are 20p coins. George takes 3 coins at random from the box. Work out the probability that he takes exactly

11 Q1 Use set notation to describe the shaded area in each Venn diagram. b) c) d) Q2 For the set A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}: Q3 A = {multiples of 2} AA BB = {4} AA BB = {1, 2,4, 6, 8, 9, 10) Draw a Venn diagram for this information. A card is selected at random from a pack of 52 playing cards. Let K be the event that the card is a king and D the event that the card is a diamond. Complete the Venn diagram. b) Find each of these: i) PP(KK DD) ii) PP(KK DD) iii) PP(KK ) iv) PP(KK DD) Q4 X and Y are two events and P(X) = 0.6, P(Y) = 0.7 and P(X Y) = 0.9. Find i) PP(XX YY) ii) PP(XX ) iii) PP(XX YY) iv) PP(XX YY) 11 Union and Intersection Set Notation LE

12 Probability Scale Q1. Impossible b) Evens c) Certain d) Likely/Unlikely depending on the time of year. e) Likely f) Impossible g) Unlikely Q2. 12 Solutions LE

13 Writing Probabilities Q1. P (Red) = d) P (Red or purple) = Q2.. P (5 or 6) = d) P (Greater than 7) = Q3. b) P (Blue) = c) P (Purple) = e) P (Black) =0 f) P (Not Black) =1 b) P (Odd number) = e) P (Factor of 12) = Spinner B b) Spinner A c) Spinner A d) Spinner B Q4. LE P(4) = d) P(>2) = Q5. P (Blue) = 13 c) P (Less than 6) = f) P (Multiple of 3) = b) P(Even) = c) P(8) =0 e) P(<7) =1 f) P(0) =0

14 Expected Frequency Q1. i) P (3) = b) Landing on 4 = 50 times c) Landing on even = 60 times ii) P (1) = d) Landing on prime = 60 times e) Landing on 6 = 40 times Q2. Shaded section = 120 times b) Even number = 120 times c) Number 3 = 40 times d) Square number = 120 times e) Prime and Shaded= 40 times Q3. iii) P (Prime) = LE Red = b) Green = c) Blue = iv) P (<6) = d) White = 0 b) Red = 100 times Green = 125 times Blue = 275 times c) Red = 200 times Green = 250 times Blue = 550 times Q4 70 times b) 90 times Q5. Profit of

15 Sample Space Diagrams Q1. HH, HT, TH, TT b) P (HH) = Q2. A1, A2, B3, B4, C5, C6, b) i) P (Number 2) = Q3. iii) P(3 and B) = v) P (C and 6) = H1, H2, H3, H4, H5, H6 T1, T2, T3, T4, T5, T6 d) ii) P (C) = iv) P (5 and A) = 0 vi) P (Not B or 3) = b) LE Q4. b) Q5. i) iv) 0 b) i) iii) 15 e) ii) ii) 0 v) iv) 1 c) P (HT or TH) = c) f) iii) vi)

16 Mutually Exclusive Events Q1. P (not 5) = Q2. P (Losing Raffle) = 85% Q3. P (Black ball) = 0.45 Q4. Sum of probabilities not equal to 100%. b) P (red) = 15% c) P (Black or Orange or Red) = 50% d) P (Not Blue) = 70% Q5. P (Not Pink) = 0.98 b) P (Pink) = 0.02 c) P (Not Green) = 0.85 d) P (Not Red) = 0.72 Q6. LE P (Red) = P (Green) = 0.12 b) 36 Reds. 16

17 Probability Trees Q1. A) b) P (R&R) = Q2. c) P (R&B or B&R) = d) P (not R) = b) P (On time & Late) = 12.75% c) P (at least one late) = 27.75% d) P (Late& Late) = 2.25% e) P ( neither late) = 72.25% Q3. P (S& S) = 0.36 b) P (S& G) = 0.24 c) P (not S) = 0.16 d) P ( S at least once) = 0.16 Q4. LE P (all three are girls) = c) P (at least two boys) = e) P (at least one girl) = 17 b) P (one boy and two girls)= d) P (no girls) = e) P (at least one boy and one girl) =

18 Solutions Q1. x = 8 b) x = 10 c) x = 7 Q2. b) 24 c) Q3. b) 24 % c) 29% Q4. b) i) ii) iii) Venn Diagrams LE

19 Solutions Q1. Q2. Q3. P (both are red) =!! c) P (only one is green) =!! Q4. P (Failure at second attempt) = 7% Conditional Probability P (both balls will be white) =! b) P (one ball of each colour is removed) =!! P (both red) =!!" b) P (one of each colour) =!!" c) P (both the same colour) =!" d) P (at least one blue) =!! Q5. P (person over the age of 60 will get the flu) = 26.25% Q6. LE Q7. P (two sandwiches of the same type) =!" Q8. P ( 2.20) =!"!" 19 P (not the same type) =!"!!!"!"!" b) P (none are red) =!! d) P (at least one is green) =!!!"

20 Q1 PP(AA BB) b) PP(AA BB) c) PP (AA BB) d) PP (AA BB) Q2 Q3 20 b) i) PP(KK DD) = 1 52 Q4 Union and Intersection Set Notation LE ii) PP(KK DD) = 4 13 iii) PP(KK ) = 12 i) PP(XX YY) = 0.4 ii) PP(XX ) = 0.4 iii) PP(XX YY) = 0.8 iv) PP(XX YY) = iv) PP(KK DD) = 3 13

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