# This unit will help you work out probability and use experimental probability and frequency trees. Key points

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1 Get started Probability This unit will help you work out probability and use experimental probability and frequency trees. AO Fluency check There are 0 marbles in a bag. 9 of the marbles are red, 7 are green and the rest are blue. Write down the fraction of the marbles that are green. In a school, 70% of the students live in the catchment area. Write down the percentage of students who do not live in the catchment area. Number sense Work out a 0.6 = b 0. = c Key points 4 = d = Probability means the chance of an event happening. The probability of an outcome = number of ways the outcome can happen total number of possible outcomes These skills boosts will help you to use the probability scale, understand mutually exclusive outcomes, predict the number of successes for an experiment, and draw and use frequency trees. The probability scale Mutually exclusive outcomes for one event Estimating successes 4 Mutually exclusive outcomes for two events and frequency trees You might have already done some work on probability. Before starting the first skills boost, rate your confidence using each concept. The probability that it will rain tomorrow 4 is. What is the 7 probability that it will not rain tomorrow? 4 A bag contains 4 red, blue and yellow marbles. Work out the probability of randomly picking a blue marble from the bag. The probability that a spinner lands on an even number is. The spinner is spun 600 times. Work out how many times the spinner is expected to land on an even number. A new car is sold in black, red or silver, and in three different models: saloon, estate and hatchback. List all possible combinations for the new car. How confident are you? 8 Unit Probability

2 The probability scale The probability scale is labelled from 0 to and is used to position and compare probabilities. The sum of the probabilities of mutually exclusive outcomes is. Guided practice The probability of this spinner landing on is. On a probability scale, mark with a cross the probability of the spinner a landing on b not landing on. Worked exam question a P() = Divide the probability scale into thirds to see where to put your cross. 0 b P() = Use the fact that the probabilities of mutually exclusive outcomes add up to. P(not ) = P() = = Mark the calculated probability on the probability scale. 0 P() means the probability of a. Use your ruler to measure the scale and then divide it by. Mutually exclusive events are outcomes that cannot happen at the same time. Mark with a cross the probability of each of these events. a The probability that it will rain tomorrow is 60%. Hint Convert the percentage b The probability that a biased dice lands on is 0.. to a decimal. c The probability of winning a tennis match is 0.7. d The probability that a day of the week selected at random contains the letter y The probability of picking a red counter out of a bag is 0.. Work out the probability of picking a counter that is not red. Hint P(not red) = P(red) = 0. The probability of a spinner landing on red is 4 9. Work out the probability of the spinner not landing on red. 4 Some male and some female students are attending tennis coaching. The probability of picking a male at random for a match is. What is the probability of picking a female? ( mark) Reflect When you know the probability of an event happening, how do you work out the probability of the event not happening? Unit Probability 9

3 Mutually exclusive outcomes for one event Mutually exclusive events are events that cannot happen at the same time. For example, getting a head or tail when flipping a coin are mutually exclusive outcomes because the coin can only land on one or the other. Getting a queen or a heart when dealing a pack of cards are not mutually exclusive outcomes because the card could be both outcomes, the queen of hearts. Guided practice A letter is picked at random from the words MUTUALLY EXCLUSIVE. Work out the probability that the letter is a vowel. Worked exam question Number of ways of getting a vowel = Number of possible outcomes = Work out the probability. Probability of a vowel = 7 7 Number of possible outcomes = number of letters number of vowels Probability of a vowel = total number of letters Write whether each pair of outcomes are mutually exclusive or not. a Spinning a -sided spinner, labelled A to E, the result is a vowel or a consonant. b Dealing an ace or a diamond from a pack of cards. c Rolling an even or an odd number with a 6-sided dice. Hint Mutually exclusive events are outcomes that cannot happen at the same time. Here are 0 letters. A A B C C C D E E F Amir takes a letter at random. a Write down the probability that he takes a letter C. b Write down the probability that he does not take a letter C. A bag contains 7 white beads and red beads. A bead is taken at random from the bag. a Write down the probability that the bead is white. b Work out the probability that the bead is red. 4 There are 0 cars in a college car park. Teachers own 7 of the cars. Students own 7 of the cars. One of the cars in the car park is picked at random. a Write the probability that a teacher owns this car. b Work out the probability that the car is not owned by either a teacher or a student. ( mark) ( marks) Reflect What is the total of the probabilities of all the mutually exclusive outcomes to an event? Use your answers to Q to help you. 0 Unit Probability

4 Estimating successes To estimate the number of successes, multiply the probability of a success by the number of trials. Guided practice The probability of a spinner landing on red is, on blue is 0. and on green is 0%. Callum spins the spinner 00 times. Estimate the number of times the spinner lands on a red b blue c green. a Multiply the probability of landing on red by the number of spins. Estimate for the spinner landing on red = 00 = times b Estimate for the spinner landing on blue = = times c Estimate for the spinner landing on green = 0% of 00 = times 00 = of Use a bar model. 0% Here is a 4-sided biased spinner. The table shows the probability for each outcome. Number 4 Probability The spinner is spun 00 times. Estimate the number of times the spinner lands on. Erin has a biased coin. She flips the coin once. The probability of getting heads is 0.. a Work out the probability of getting tails. Jamal flips this coin 00 times. b Estimate the number of tails Jamal gets. Toby plays a game. The probability that Toby wins the game is %. Toby plays the game 40 times. Work out an estimate for the number of times he wins the game. 4 An ordinary 6-sided dice is rolled in a game. Bonus points are scored for rolling a 6. During the game, the dice is rolled 0 times. Estimate the number of times the dice lands on 6. Oti takes a counter from a bag at random. The probability that she takes a red counter is 40%. Oti writes down the colour of the counter and returns it to the bag. Oti does this 0 times. Work out an estimate for the number of times that Oti takes a red counter from the bag. ( marks) 4 Reflect Explain how you know that the spinner in Q is biased. Unit Probability

5 4 Mutually exclusive outcomes for two events and frequency trees When listing outcomes for more than one event, be systematic. You can use sample space diagrams and two-way tables to list outcomes for two or more events. Frequency trees show the number of options for different outcomes. Guided practice Hasan flips two coins once. Each coin will land on either heads or tails. a List all the possible outcomes that Hasan could get. b Work out the probability that Hasan gets two tails. Worked exam question a Write heads for the first coin with all possible outcomes on the second coin. Coin Coin heads heads heads Now write tails for the first coin with all possible outcomes on the second coin. Be systematic with your list to help cover all possible outcomes. b Write the number of possible outcomes. Number of possible outcomes = Probability of two tails = Count the number of possible outcomes in part a. Probability of an outcome = number of ways the outcome can happen total number of possible outcomes A café sells a soup and sandwich on a special offer. Customers can choose tomato or vegetable soup and a ham or cheese sandwich. List all possible combinations of the soup and sandwich. Six students stand for school council. Three of the students are boys: Alfie, Tom and George. Three of the students are girls: Lexi, Clara and Esme. One boy and one girl are picked at random. List all possible outcomes for the school council. Here are two -sided fair spinners. Zainub is going to spin each spinner once. Her score is the sum of the two numbers. Spinner A Spinner B Unit Probability

6 a Complete the sample space diagram for each possible score. Spinner A 4 Spinner B b Work out the probability that Zainub gets i a score of 4 ii a score greater than. 4 A mobile phone company surveys 00 of its customers. 6 of these customers have a smartphone. 4 of the 00 customers have a contract. 7 of the other mobile phone customers use pay as you go. a Use this information to complete the frequency tree. One of the smartphone customers is selected at random. b Work out the probability that this smartphone customer has a contract. c Complete the two-way table using the information given in this question. Contract Pay as you go Total 00 Smartphone Other mobile phone Total smartphone other mobile phone contract pay as you go contract pay as you go Hint Put the numbers you are given into the correct cells in the table. Next work out missing numbers in rows and columns where you have two values. 0 students had some homework to do. of the students were boys. of the 0 students did not do their homework. 6 of the girls did do their homework. 0 boys girls did do homework did not do homework did do homework did not do homework a Use this information to complete the frequency tree. One of the girls is selected at random. b Work out the probability that this girl did not do her homework. ( marks) ( marks) Reflect In Q4, did you find the frequency tree or the two-way table easier for displaying the data? Why? Unit Probability

7 Get back on track Practise the methods Answer this question to check where to start. Check up Number 4 The table shows the probabilities that a -sided spinner lands on each outcome. Probability Tick the correct probability for the spinner landing on 4. A B C D If you ticked C go to Q. If you ticked A, B or D go to Q for more practice. Work out a 0. = b 0.6 = c 0. = d 0.8 = The probability that a spinner lands on red is 0.4. Work out the probability that it does not land on red. A packet of sweets contains orange and 7 red sweets. A sweet is taken from the packet at random. a Write down the probability that the sweet is orange. b Work out the probability that the sweet is red. 4 In a game, there are cards with the numbers to 4. The table shows the probability of getting each number. Number 4 Probability a Work out the probability of getting a 4. b The cards are shuffled 60 times during the game and each time a card is dealt. Estimate the number of times a is dealt. Taylor takes a letter tile from a bag at random. The probability that she takes a vowel is 0%. Taylor writes down the letter on the tile. She then puts the tile back in the bag. Taylor does this 80 times. Estimate the number of times that Taylor takes a vowel. 6 Here is the menu in a café. Sally is having a meal in the café. She chooses one starter and one main course. List all the different meals Sally can choose. Starter Melon Soup Menu Main course Pasta Chicken Fish ( marks) 4 Unit Probability

8 Get back on track Problem-solve! s There are 0 sweets in a box. x sweets are red. The rest of the sweets are yellow. Tracy takes a sweet from the box at random. Write down an expression, in terms of x, for the probability that Tracy takes a yellow sweet. ( marks) Kate has a 4-sided spinner. Number 4 The sides of the spinner are numbered,, and 4. Probability 0. x 0. The spinner is biased. The table shows the probability that the spinner lands on, or 4. The probability that the spinner lands on is x. a Find an expression, in terms of x, for the probability that the spinner lands on. Give your answer in its simplest form. Kate spins the spinner 00 times. b Write down an expression, in terms of x, for the number of times the spinner is likely to land on. ( marks) ( mark) A restaurant sold 80 pizzas last week. 0 of the pizzas had a thin crust base. 49 of the deep pan bases had a meat topping. 64 of all the pizzas had a vegetarian topping. a Use this information to complete the frequency tree. ( marks) One of the customers who chose a deep pan base is selected at random. b Work out the probability that this customer had a meat topping on their pizza. 80 deep pan thin crust vegetarian meat vegetarian meat ( marks) 4 Lei has a biased coin. When she flips the coin once, the probability of getting heads is x. a Write down an expression, in terms of x, for the probability of getting tails. Lei flips the coin 00 times. b Write down an expression, in terms of x, for an estimate of the number of times she gets heads. ( mark) ( marks) Now that you have completed this unit, how confident do you feel? 4 The probability scale Mutually exclusive outcomes for one event Estimating successes Mutually exclusive outcomes for two events and frequency trees Unit Probability

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