SOLUTIONS Basic ideas to start 3) The complete text is: Imagine you warm water up to 100ºC. You know what is going to happen: the water will boil. And you can be sure about it. But if you cannot see your five best friends at all (and you don`t know where they are)… you cannot know the result of your choice. When you know for sure the result of an experiment it is a determinist experiment. When you know the possible results or outcomes of an experiment, but you cannot know which of them will occur, it is a random experiment. Things can happen: Classify these experiments and enumerate the possible results: You cool water up to 0ºC. Classification: it is a deterministic experiment. Possible results: it freezes. You toss a coin. Classification: it is a random experiment. Possible results: head or tail. You pick a card. Classification: it is a random experiment. Possible results: every card of the deck You are vaccinated. Classification: it is a Possible results: you won’t deterministic experiment. suffer the illness. 4) Rolling a die: a) Sample space: {1,2,3,4,5,6} b) {1,3,5} c) {2,4,6,} d) The text is… The set of the outcomes of a random experiment is called sample space. Material
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An event is a subset of the sample space. Probability is… 1) What event is more likely to happen? a) Possible answer: I choose “obtaining an odd number" because the number of events. b) A C B D c) The right option: You cannot know what will happen. 2) Tossing the coin: b) You have to divide the number of times you obtained “head” by 20 (or the number of times you tossed the coin). You have to do the same for “tail”. c) Expected kind of answer: I think I will obtain “head” half the times, 450 times, and “tail” half the times, 450 times too. d) Expected kind of answer: I think they have the same probability to occur. 3) About relative frequencies: a) Relative frequencies: 0.35, 0.4, 0.458, 0.502, 0.50001. Conclusion: the more you repeat the experiment, the more the relative frequency gets close to 0.5. The probability of the event seems to be 0.5. 4) The true options are d) and g). The rest are false. 5) The text is: The probability of an event in a random experiment is the relation between the number of times the event occurs and the number of times the experiment is repeated. The probability of an event gives information about “how likely the event is to happen”. Probability is always expressed by a number between 0 and 1 (inclusive). Measuring probability 1) The probabilities are: 4
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To drinking water this week: 1 •
To having a negative number of brothers or sisters: 0 •
To winning 1.23 tickets for a concert: 0 •
To visiting ten continents: 0 Material
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To sleeping at least two hours this month: 1 •
To seeing the rain once in your life if you live in La Coruña: 1 • To being born on the 35th of May: 0 2) Opposite: a) Not to play Not to be • To obtain an odd number when you throw a die. •
To o btain tail when you toss a coin. • To be vaccinated. Eight • To sleep three hours. • To play some sport. Even Head b) Three possible sentences: • Not to be vaccinated • To obtain an even number when you throw a die. • Not to play any sport. 3) There are several options, for example: if you go to class you don’t stay at home. 4) Same probability: • To throw a die. (The six possible results have the same probability if the die is normal). • To pick a card. . (The possible results have the same probability if the deck of cards is normal). • To select a book in a library. (The possible results have different probability if we consider we usually select books we like). • To pick a ball. (Depends on the homogeneity of the balls). • To select a cinema. (The probability of the possible results will depend on the kind of movie, location of every cinema…) • To choose a color. (The possible results have the same probability if there is no preference about the colors). Material
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5) Complete text: As you know the probability of an event of a sample space is a decimal number between 0 and 1 (ALWAYS). Imagine a random experiment with a sample space where:  There are finite possible outcomes.  Every outcome has the same probability to occur.  The outcomes are incompatible events. To calculate the probability of an event of an experiment in these conditions, we can use Laplace law. 6) Using Laplace law: a) There are… 3 blue t-­‐shirts. 2 yellow t-­‐shirts. 1 striped t-­‐shirts. 4 purple t-­‐shirts. b) The fractions are… Color of the t-­‐shirts blue yellow striped purple fraction 3/10 2/10 1/10 4/10 c) The probabilities are: 0.1, 0.9, 0.3, 0.8 7) The solutions are: a) 2/6= 0.33… b) 4/40=0.1 c) 6/15=0.4 d) 1-­‐0.4=0.6 8) True or false: • True, they both complete the total sample space. • False, their probability can vary very much. • True, that is the definition of equiprobable events. • False: the probability can be calculated subtracting the probability of A to 1. 6
Material
de ESO: Probability
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MaterialAICLE
AICLE 2ºSecundaria
2º E.S.O. : ANCIENT
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