WHAT IS PROBABILITY?

CLIL projectClass II C

ACTIVITIES

CLIL projectClass II C

likely probablecoin outcome

sample spaceJack King

probabilityset event

Gambing

KEY WORDS 1. The Puzzle Find the words and eliminate the corresponding letters from the puzzle. The remaining letters form the name of the French mathematician who first proposed the classical definition of probability.

D O L O O H C S L K G S R I R U M J A C K A C N O A R N A T N I O C C I I Y C U S D W C G D E S P L L P C S S R I O N L A E L E R C A E O A N M I P I O K O O P L T T C G E K D R I B P R O B A B I L I T Y L A E S A C A S T N E V E C B T E S T G A M B L I N G L E E C A P S E L P M A S E

2. Connect an English word with its Italian translation

CARD SCEGLIERE A CASO COIN GETTARE (i dadi) DIE GIOCO D’AZZARDO

to DRAW RISULTATO EVENT DADO

GAMBLING CAPITARE JACK EVENTO KING SPAZIO CAMPIONARIO

LIKELY PROVARE to OCCUR CARTA DA GIOCO

OUTCOME GETTARE (la moneta) to PICK PROBABILITÀ

PROBABILITY INSIEME to ROLL RE

SAMPLE SPACE PRENDERE SET MONETA

TEST PROBABILE to TOSS FANTE

3. Put in the spaces the correct words from the list of key words.

a) The word _____________, ____________, and hazardous are synonyms. b) Tossing a ___________ I got head. Head is the _____________ of my experiment. c) All the possible outcomes that can occur when I execute the experiment form the ______________. d) The _________, the Queen and the ____________ are three cards of a pack. e) The measure of how likely an event is, is called ________________. f) The particular outcome or __________ of outcomes we are interested in, is an _________. g) ____________ is dangerous for our pocket!

CAR DIE JACK OUTCOME PROBABLE SET CARD DRAWING KING PASCAL ROLLING TEST CASE EVENTS LIKELY PICK SAMPLE SPACE TOSS COIN GAMBLING OCCUR PROBABILITY SCHOOL

Name and Surname ___________________

ACTI VI TY 2: PROBABI LI TY

1. There are three possible ways to find probability: LACISCASL EINFTIDON, QFIUSTTNEERU EFDOTNINI, ECJTIUVBSE IBLIARTBYOP.

__________ __________, ___________ ___________, _____________ ___________. 2. Write a phrase about Bruno De Finetti.

______________________________________________________________________ ______________________________________________________________________

3. Complete the formula for the classical definition of Probability: Probability Of An Event P(A) = The Number of __________________cases The number of cases

4. The ________________ of event A is the number of ways event A can occur ___________ by the

total number of possible ___________. The probability of event A is the number of ____________ cases (outcomes) divided by the total number of possible ______________ (outcomes).

cases divided f avourable outcomes probability

5. Exercises:

a) What is the probability of rolling an even number with a die? b) In a bag we have 10 pens: 3 green, 4 red and the remaining blue. Picking a pen up, what is

the probability of picking a green one? What is the probability for a red one? c) Playing Tombola, what is the probability of drawing a multiple of 7? d) In a box we have 5 candles, three used and two new ones. What is the probability of drawing

a new one from the box?

6. Which of the following events are certain and which impossible? In the last column, write the appropriate probability (zero or one). What is the probability that…

EVENT certain impossible P(A)

…rolling a die, a number greater than 8 is drawn? … in a this year Formula 1 race, Alex Del Piero wins? … playing tombola, a number less than 100 is drawn? … in the today Mathematics test, Lisa receives 12 (she’s is very good, indeed!)? … Giovanni works in Nolandia (?)? … rolling two dice, the sum of the numbers is 20? … choosing a letter from the alphabet, it’s a consonant or a vowel. … picking a card up from an ordinary pack, it’s a 15 of clubs. … choosing a boy among your schoolmates, he is a Carducci student? … this morning you don’t meet a Lunar girl?

CLASSICAL DEFINITION FREQUENTIST DEFINITION SUBJECTIVE PROBABILITY

Bruno De Finetti proposed the subjective theory of Probability and he worked in Triest in

the first half of the XX century (University, Assicurazioni Generali)

favourable

possible

probability dividedfavourableoutcomes

cases

3/6 = 1/2

3/10 4/10 12/90 =6/45

2/5

X 0

X 0

X 1

X 0

X 0

X 0

X 1

X 0

X 1

X 1

ACTIVITY 3

Crossword: the Theory of probability

1 2 3

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5 6

7 8 9

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Across:

1 It's a card with the image of a soldier

4 The probability of a certain event

6 The probability definition which involves frequency

8 In Italian it's: scegliere a caso

9 In the Laplace's formula, these are the cases at the numerator

10 Synonym of probable

11 In the coin, it's the opposite of tail

12 An event which has P(A)=1

13 The object I toss

14 The probability of an impossible event

Down:

2 In Italiano it's "caso"

3 The verb for a coin

5 The measure of how probable an event is

7 The "De", famous French mathematician

8 Famous Italian mathematician who proposed the subjective theory of probability

13 A jack is a .....

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EXERCISES

1. In a sack we have three blue little balls, two white balls and four yellow balls. Which is the probability of drawing a blue ball from the sack? Which is the probability of drawing any ball except a blue one? (use complement definition)

2. In an ordinary pack (for “briscola”) there are 40 cards. Four of those are jacks. Drawing a

card, which is the probability of getting a jack? Which is the probability of drawing any card, except a jack? (use complement definition)

3. Which is the probability of drawing from the sack a Tombola number bigger than 75? Which is the probability of drawing a number less or equal to 75? (use complement definition).

4. Rolling a die, which is the probability of rolling an odd number? And which is the probability of rolling an even number? (use complement definition)

5. I’m thinking about a day of the year. Which is the probability that it is exactly your birthday? Which is the probability that isn’t your birthday? (use complement definition).

6. Three pens are good and five are not. Drawing one pen of the set, which is the probability of drawing a good pen? Which is the probability of getting a bad one? (use complement definition).

7. If I have a probability equal to 0,23 to complete this activity in half an hour, which is the probability that I will not complete it in half an hour? (use complement definition).

8. A party has a probability P(A)=0,73 of winning the elections. Which is the probability that

this party loses? (use complement definition).

9. Which is the probability of rolling a 7 with an ordinary die? Which is the probability of rolling any number except a 7? (use complement definition).

P(blue)=3/9=1/3 P(not blue)= 1 – 1/3=2/3

P(jack)=4/40=1/10 P(not jack)= 1 – 1/10=9/10

P(>75)=15/90=1/6 P(not >75)= 1 – 1/6=5/6

P(odd)=3/6=1/2 P(even: not odd)= 1 – 1/2=1/2

P (your birthday)=1/365 P(not your birthday)= 1 – 1/365=364/365

P(good)=3/8 P(bad: not good)= 1 – 3/8=5/8

P(I complete)=0,23 P(I don’t complete)= 1 – 0,23 = 0,77

P(wins)=0,73 P(loses: doesn’t win)= 1 – 0,73 = 0,27

P(7)=0 P(not 7)= 1 – 0 = 1

Name and Surname______________________________

ACTI VI TY 4. COMPOUND EVENT

I) Answer to the following questions:

a) Which is the key conjunction in the compound event?

1) OR 2) AND 3) BECAUSE 4) THAT

b) When two events are independent?

1) When they are logically connected. 2) When the fact that A occurs affects the probability of B occurring. 3) When one follows the other. 4) When the fact that A occurs does not affect the probability of B occurring.

c) Which is the arithmetical operation for the calculation of a compound event?

1) Subtraction 2) Addition 3) Division 4) Multiplication

d) In Theory of probability the symbol P(B | A) means that

1) the probability of event B must be divided by the probability of event A 2) the probability of event B is influenced by the fact that event A has occurred 3) the probability of event B must be divided by the probability of event A 4) the probability of event B is not influenced by the fact that event A has occurred

e) If you draw a coloured ball from a sack (event A) and then you draw another one (event B),

the two events: are always independent are never independent are independent only if we put back (replace) the first ball in the sack are independent only if we don’t put back (replace) the first ball in the sack

II) In the table we have nine couples of events (event A; event B). Indicate the couples of dependent (D) events and of independent events (I)

EVENT A EVENT B I D

1 Rolling a die, number 3 is drawn Rolling the same die, number 4 is drawn x

2 Picking a card, it’s a kingPicking a card from the same pack without replacement, it’s a jack

3 Picking a card, it’s a kingPicking a card from the same pack with replacement, it’s a four

4 A man chosen randomly is a smoker A man chosen randomly is affected by heart diseases

5One of you, chosen randomly, likes English

One of you, chosen randomly, likes CLIL course

6 Number 37 is drawn, playing Tombola. Number 43 is drawn, playing Tombola

7Schumacher wins the next F1 Championship

Juventus wins 2005-2006 Italian football championship

8 Schumacher wins A Ferrari car wins

9 Number 18 is drawn at roulette Number 12 is drawn at roulette

III) Calculate the probabilities of the compound events formed by the two events on a

line in the table (lines: 1,2,3,9) (look at the example of line 1)

P(A) P(B) dependent/independent P(compound event)

LINE1: EVENT A: rolling a die, number 3 is drawn; EVENT B Rolling the same die, number 4 is drawn

1/6 1/6 independent P = 1/6 1/6 = 1/36

LINE2:

LINE3:

LINE9: [in roulette there are 37 numbers]

x

x

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x

4/52 4/51 dependent P=4/52 4/51=16/2652

4/52 4/52 independent P=4/52 4/52=16/2704

1/37 1/37 independent P=1/37 1/37=1/1369

Name and Surname______________________________

ACTI VI TY 5. MUTUALLY EXCLUSI VE EVENTS

Crossword

Across: 1. Two events, A and B, are _____ if the fact that A

occurs does not affect the probability of B occurring. 2. The "rule" for compound event. 3. The "key" conjunction for two independent events

when I ask that one of them occurs 6. Two events are mutually ____ if it is impossible for

them to occur together. 7. The "rule" for finding the probability in the "OR"

case. 8. The "key" conjunction for the compound event 9. ____’s diagrams show the sets we are studying

Down: 1. The operation between sets which involves the

compound event. Its symbol is . 4. Two events are _________ exclusive , if it is

impossible for them to occur together. 5. We say "mutually exclusive" events or _______

events

II) In the table we have nine couples of

events (event A; event B). Indicate the couples of mutually exclusive (disjoint D) events and of not mutually exclusive (NME)

EVENT A EVENT B ME NME 1 Number 32 is drawn, playing Tombola Number 87 is drawn, playing Tombola x 2 Rolling a die, a 3 is drawn Rolling a die, a 4 is drawn 3 Drawing a cards it’s a club Drawing a card, it’s a king 4 Schumacher wins Alonso wins 5 Schumacher wins A Ferrari car wins 6 Choosing randomly a letter, it’s a vowel Choosing randomly a letter, it’s a consonant 7 Rolling a die an odd number is drawn Rolling a die a multiple of 3 is drawn 8 Choosing randomly a day, it’s in March Choosing randomly a date, it’s an odd number 9 Choosing randomly a song, it’s sung in English Choosing randomly a song, it’s sung by an American singer

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III) Find the probability of the events formed by the following couples of disjoint events:

Event A: “Rolling a die, a 3 is drawn” P(A) = _______

Event B: “Rolling a die, a 5 is drawn”P(B) = _______

Which is the probability of rolling a 3 OR a 5 on a die?

Event A: “Drawing a card it’s a spade” P(A) = _______

Event B: “Drawing a card it’s a club”P(B) = _______

Which is the probability of drawing a spade OR a club from an ordinary deck?

Event A: “Getting number 31 at the roulette” P(A) = _______

Event B: “Getting number 37 at the roulette”P(B) = _______

Which is the probability of getting a 31 OR a 37 at the roulette?

Event A: “Drawing a Tombola number, it’s a multiple of 13” P(A) = _______Event B: “Drawing a Tombola number, it’s a multiple of 9” P(B) =

_______Which is the probability of drawing a multiple of 13 OR a multiple of 9?

Event A: “the probability that tomorrow it’ll be cloudy is 0,45”Event B: “the probability that tomorrow it’ll be snowy is 0,15”Which is the probability that tomorrow it will be cloudy OR snowy?

In a jar there are 14 coloured marbles: 6 green, 3 red, 5 blue ones.Event A: “A green marble is drawn” P(A) = _______Event B: “A blue marble is drawn” P(B) = _______Which is the probability that a green OR a blue marble is drawn?

1/6

1/61/6 + 1/6 =2/6

1/4

1/41/4 + 1/4 =1/2

1/37

0/371/37 + 0/37 =1/37

7/90

9/907/90 + 9/90 =16/90

0,45+0,15 =0,60

6/145/14

6/14 + 5/14 =11/14

Name and Surname______________________________

ACTIVITY 6. TOTAL PROBABILITY and REVIEW

I)Find the probability of the events formed by the following couples of not disjoint events:

Event A: “Rolling a die, a multiple of 2 is drawn” P(A) = 3/6Event B: “Rolling a die, a multiple of 3 is drawn” P(B) = 2/6Event (AB): “Rolling a die, a multiple of 2 and a multiple of 3 is drawn” P (AB) = 1/6Which is the probability of rolling a multiple of 2 OR a multiple of 3 on a die?

P (AB)=3/6 + 2/6 – 1/6 = 4/6

Event A: “Drawing a card it’s a heart” P(A) = _______Event B: “Drawing a card it’s a face” P(B) = _______Event (AB): “Drawing a card, it’s a heart and a face” P (AB) = _______Which is the probability of drawing a heart OR a face from an ordinary deck? P (AB)=

Event A: “Drawing a Tombola number, it’s a multiple of 12” P(A) = _______Event B: “Drawing a Tombola number, it’s a multiple of 10” P(B) = _______ Event (AB): “Drawing a number, it’s a multiple of 12 OR a multiple of 10”

P (AB) = _______Which is the probability of drawing a multiple of 12 OR a multiple of 10?

P (AB)=

In a jar there are 25 coloured marbles: 6 green and big, 8 green and small, 5 blue and big, 6 blue and small.

Event A: “A green marble is drawn” P(A) = _______Event B: “A big marble is drawn” P(B) = _______ Event (AB): “Drawing a marble, it’s green and big” P (AB) = _______Which is the probability that a green OR a big marble is drawn?

P (AB)=

13/52

12/523/52

13/52+12/52-3/52

7/909/90

1/90

7/90+9/90-1/90=15/90

14/25

11/256/25

14/25+11/25-6/25=19/25

II) Solve the following exercises (repetition): 1. What is the probability that choosing randomly among the students of your own class for a

test, the chosen student is exactly you? 2. What is the probability that in this year Formula 1 race, Alex Del Piero wins?

3. What is the probability that choosing a boy among your schoolmates, he is a Carducci student?

4. In a sack we have three blue little balls, two white balls and four yellow balls. Which is the probability of drawing a blue ball from the sack? Which is the probability of drawing any ball except a blue one? (use complement definition)

5. I’m thinking about a day of the year. Which is the probability that it is exactly your birthday? Which is the probability that isn’t your birthday? (use complement definition).

6. What is the probability that, picking a card, it’s a king and afterwards, picking another card from the same deck with replacement, it’s a four?

7. Playing Tombola number 33 is drawn and, afterwards, number 12 is drawn. Which is the probability of this compound event?

8. Which is the probability of rolling a 3 OR a 5 on a die?

9. If the probability that tomorrow it’ll be cloudy is 0,45 and the probability that tomorrow it’ll be snowy is 0,15, which is the probability that tomorrow it will be cloudy or snowy?

10. Playing Tombola, which is the probability of drawing a multiple of 12 or a multiple of 10?