Mathematics 10 D 2 Probability Mathematics Proper Thinking

Mathematics 10 D 2 Probability ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

The language of probability ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

The language of probability ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Find Outcomes of simple events • For Simple Events – count the outcomes • Examples: One Die- 6 outcomes One coin- 2 outcomes One deck of cards- 52 outcomes One fair number cube- 6 outcomes ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

sample space If the outcomes of an event are equally likely then we can calculate the Sample space: For example, a bag contains 1 yellow, 3 green, 4 blue and 2 red marbles. This spinner has 8 equal divisions. For example, ……………… ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Sample space diagrams A coin and a ten-sided dice are thrown and the outcomes recorded in the two-way table below. Dice Coin 1 2 3 4 5 6 7 8 9 10 H T 1) How many outcomes are there? Make up a fair game and an unfair game based on scoring points from outcomes in the table. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Sample space diagrams A coin and a ten-sided dice are thrown and the outcomes recorded in the two-way table below. Dice Coin 1 2 3 4 5 6 7 8 9 10 H 1, H 2, H 3, H 4, H 5, H 6, H 7, H 8, H 9, H 10, H T 1, T 2, T 3, T 4, T 5, T 6, T 7, T 8, T 9, T 10, T ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Outcomes and events An event can have several outcomes. What are the possible outcomes when throwing a ten-sided dice? If a dice has ten faces, then there are ten possible outcomes, one for each face of the dice. Can you think of an event that has two outcomes? A simple example of an event that has two outcomes is flipping a coin. The two outcomes are heads and tails. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Outcomes and events Each outcome of a given event has a probability or a chance of occurring. What are the chances of each outcome from throwing a ten-sided dice? Assuming that the dice is fair, the chances of each outcome 1 occurring is 10. Can you think of an event that has two outcomes which have probabilities that are not equal? One example is that a randomly chosen person will be rightor left-handed. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Two outcomes In which of these scenarios does each outcome have an equal chance of occurring? It will rain tomorrow. A child born will be a boy. A coin will show tails when it is flipped. A number selected at random from 1 to 100 will be even. When a dice is thrown, it will show a square number. The next person to walk into the room will be right handed. The bus will be on time tonight. The bus driver will be female. When a dice is thrown, it will show a prime number. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

The probability scale Probability is measured on a scale from 0 to 1. 1 0 impossible 2 even chance If an event is impossible or has no chance of occurring, then it has a probability of 0. If an event has an even chance of occurring, it has a probability of 0. 5. 1 certain If an event is certain it has a probability of 1. Probabilities can be written as fractions, decimals, ratios and percentages. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

The probability scale We measure probability on a scale from 0 to 1. If an event is impossible or has no chance of occurring, then it has a probability of 0. If an event is certain it has a probability of 1. This can be shown on the probability scale as: 0 impossible 1 2 even chance 1 certain Probabilities are written as fractions, decimals and, less often, as percentages. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Using the probability scale ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Calculating probability If the outcomes of an event are equally likely then we can calculate the probability using the formula: Probability of an event = Number of possible successful outcomes Total number of possible outcomes For example, a bag contains 1 yellow, 3 green, 4 blue and 2 red marbles. What is the probability of pulling a green marble from the bag without looking? P(green) = 3 10 or 0. 3 or 30% ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Calculating probability This spinner has 8 equal divisions: What is the probability of the spinner a) landing on a blue sector? b) landing on a red or green sector? c) not landing on a green sector? a) P(blue) = 1 8 6 3 = 8 4 4 1 c) P(not green) = = 8 2 b) P(red or green) = ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Calculating probabilities with spinners ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Calculating probabilities The probability of a spinner landing on yellow is 0. 2. What is the probability of not landing on yellow? 1 – 0. 2 = 0. 8 If the probability of an event occurring is p then the probability of it not occurring is 1 – p. A spinner has green, red and blue sections. Landing on red is twice as likely as landing on green. Fill in the missing probabilities: Green 0. 26 Red 0. 52 Blue 0. 22 ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Tree diagrams You have a pack of fifteen playing cards. Six of the cards are red and the rest are black. Write down the decimal probability of choosing a red card at random from the pack. P(red card) = 6 15 = 0. 4 You decide to pick a card at random from the pack, replace it and then pick another. If you pick two red cards, you will stay in and do your homework. If you don’t, you will go out to a party. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Tree diagrams All of the possible outcomes can be shown using a probability tree diagram: 1 st card red 0. 4 0. 6 black 0. 4 0. 6 2 nd card red Probabilities P(R, R) = 0. 4 × 0. 4 = 0. 16 black P(R, B) = 0. 4 × 0. 6 = 0. 24 red P(B, R) = 0. 6 × 0. 4 = 0. 24 black P(B, B) = 0. 6 × 0. 6 = 0. 36 ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Tree diagrams Probabilities P(R, R) = 0. 4 × 0. 4 = 0. 16 P(R, B) = 0. 4 × 0. 6 = 0. 24 P(B, R) = 0. 6 × 0. 4 = 0. 24 P(B, B) = 0. 6 × 0. 6 = 0. 36 1) What do the probabilities add up to? 0. 16 + 0. 24 + 0. 36 = 1 2) What are the chances of your doing your homework? P(R, R) = 0. 4 × 0. 4 = 0. 16 3) What are the chances of going to the party? P(not R, R) = 1 – 0. 16 = 0. 84 ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Combined events: horse race You are going to take part in a simulation of a horse race. There are twelve horses, numbered 1 to 12. 1 2 3 4 5 6 7 8 9 10 11 12 Throw two dice and add the numbers together. Each total represents a horse. Fill in a square in the table each time the horse’s number comes up. This represents the horse moving forward one place. The first horse to reach the top wins the race. Before you start, place a bet on a horse to win. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Horse race Play the game a few times. You can bet on different horses each time, or stick with the same horse. Discuss these questions with a partner: 1) Which horses won in your races? 2) 3) 4) 5) 6) 7) Which horses are unlikely to win? Explain. Are there any horses you wouldn’t bet on? If so, why? Which horses are the best ones to bet on? Why are some horses more likely to win than others? How would the game change if you used ten-sided dice? To get a total of 8, you can throw a 3 and a 5 or a 5 and a 3. What other combinations produce a total of 8? 8) How many combinations produce a total of 12? ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Horse race evaluation Horse Combinations 1 Ways Probability 0 0 2 1, 1 1 3 1, 2 2, 1 2 4 1, 3 3, 1 2, 2 3 5 1, 4 4, 1 2, 3 3, 2 4 6 1, 5 5, 1 2, 4 4, 2 3, 3 5 7 1, 6 6, 1 2, 5 5, 2 3, 4 4, 3 6 8 2, 6 6, 2 3, 5 5, 3 4, 4 5 9 3, 6 4, 5 5, 4 4 10 4, 6 5, 5 3 11 5, 6 6, 5 2 12 6, 6 1 1 36 2 36 3 36 4 36 5 36 6 36 5 36 4 36 3 36 2 36 1 36 = = = 1 18 1 12 1 9 = 1 6 = = = 1 9 1 12 1 18 ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Sample space diagrams This table is another way of displaying all the outcomes from throwing two dice and adding them together. It is called a sample space diagram. Second die + 1 1 2 3 4 5 3 First die 2 3 8 4 Fill in the rest of the cells in the table. Colour in all the twos one colour, the threes another colour etc. What patterns do you notice in the table? 5 6 6 9 ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Sample space diagrams Second die + First die 1 2 3 4 5 6 1) What is the probability of 1 2 3 4 5 6 getting a total more than 5? 2 3 4 5 6 7 2) What is the probability of getting a total less than 10? 3 4 5 6 7 8 3) What is the probability of 4 5 6 7 8 9 getting a total that is a square number? 5 6 7 8 9 10 4) What is the probability of 6 7 8 9 10 11 getting an even total? 7 8 9 10 11 12 5) What is the probability of getting an odd total? Now make up your 6) What is the probability of own questions! getting a total less than 13? ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Combined events with coins and dice A coin and a ten-sided die are thrown and the outcomes recorded in the two-way table below. Dice Coin + H T 1 2 2, H 3 4 5 6 7 8 9 10 7, H 8, T Complete the table to show all the possible outcomes. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Combined events Die Coin + 1 2 3 4 5 6 7 8 9 10 H 1, H 2, H 3, H 4, H 5, H 6, H 7, H 8, H 9, H 10, H T 1, T 2, T 3, T 4, T 5, T 6, T 7, T 8, T 9, T 10, T 1) How many outcomes are there? 2) What is the probability of getting a head an even number? 3) What is the probability of getting a tail and a square number? Make up a game where you win points if you get certain outcomes from the table above. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Other combined events Two four-sided dice are thrown and the numbers added together. Construct a sample space diagram to show all the outcomes. What is the probability of getting: First die Second die + 1 2 3 4 1) a total more than 4? 1 2 3 4 5 2) a total less than 8? 2 3 4 5 6 7 8 3) a prime number total? 4) a total that is at least 3? 5) a total of 4 or 5? 6) the same number on both dice? 7) a lower number on the first dice? ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Calculating the number of outcomes ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Using fractions to find probabilities First die Second die + 1 2 3 4 5 6 7 4 5 6 7 8 Use the table to find the probability of getting a score of 3 or 4. The probability of getting a score of 3 or 4 can be written as P(3 or 4). This is the same as the probability of getting a 3 added to the probability of getting a 4. P(3 or 4) = P(3) + P(4) = 2 16 3 + 16 = 5 16 P(5 or 6) = P(5) + P(6) = 4 16 3 + 16 = 7 16 P(2 or 7) = P(2) + P(7) = 1 16 2 + 16 = 3 16 ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Using fractions to find probabilities Two six-sided dice are thrown. + 1 2 3 4 5 6 7 8 3 4 4 5 5 6 6 7 7 8 8 9 9 10 5 6 7 8 9 10 11 12 Work out P(3 or 4) by adding fractions. P(3 or 4) = P(3) + P(4) P(5 or 6) = P(5) + P(6) = = 2 36 3 + 36 = 5 36 Work out P(5 or 6) by adding fractions. 4 36 5 + 36 = 9 36 = 1 4 ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Independent events Peter rolls an unbiased six-sided die fifty times and doesn’t roll a six once. He says “I must get a six soon!” Do you agree? Each roll of the die is unaffected by the previous outcomes. The next roll of the die is no more likely to be a six than any of the other rolls. The probability is still 61. Two events are said to be independent if the outcome of one has no effect on the outcome of the other. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Independent events First dice + 1 Second dice 2 3 4 5 What is the probability of getting a 1 followed by a 2 when two dice are thrown? 6 1 1, 2 1, 3 1, 4 1, 5 1, 6 2 2, 1 2, 2 2, 3 2, 4 2, 5 2, 6 3 3, 1 3, 2 3, 3 3, 4 3, 5 3, 6 4 4, 1 4, 2 4, 3 4, 4 4, 5 4, 6 5 5, 1 5, 2 5, 3 5, 4 5, 5 5, 6 6 6, 1 6, 2 6, 3 6, 4 6, 5 6, 6 P(1) and P(2) = P(1) × P(2) = You can use the table to find the number of successful results and the number or possible results… …or you can multiply the probabilities of the scores on each dice, P(1) and P(2): 1 6 × 1 6 = 1 36 It is important to remember that his method can only be used when the two events are independent. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Independent events First dice + 1 Second dice 2 3 4 5 6 1 1, 2 1, 3 1, 4 1, 5 1, 6 2 2, 1 2, 2 2, 3 2, 4 2, 5 2, 6 3 3, 1 3, 2 3, 3 3, 4 3, 5 3, 6 4 4, 1 4, 2 4, 3 4, 4 4, 5 4, 6 5 5, 1 5, 2 5, 3 5, 4 5, 5 5, 6 6 6, 1 6, 2 6, 3 6, 4 6, 5 6, 6 What is the probability of getting a 1 on the first die and an even number on the second die? P(1) = 1 6 P(even) = P(1) and P(even) = P(1) × P(even) = 1 6 × 1 2 Compare this with the result from the table: 1 2 = 3 36 1 12 = 1 12 ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Combining probabilities using multiplication The probability of two independent events happening at the same time is: P(A and B) = P(A) × P(B) This method applies when there are more than two events. For example: P(A and B and C) = P(A) × P(B) × P(C) ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Important Sites http: //wwwk 6. thinkcentral. com/content/hsp/math/hspmath/ca/comm on/itools_int_9780153616334_/probability. html ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation