Random phenomenonIt is one which, when repeated several times, presents unpredictable results.Ex.: When rolling a die, we can obtain any one of its faces. It is not possible to predict which one it will be.Sample spaceIt is the universal set of all possible results of the random phenomenon. It is represented by U.Ex.: When rolling a die, U = {1,2,3,4,5,6}.EventIt is any subset of the sample space.Ex.: Obtaining an even number when rolling a die. E= {2,4,6}.Complement of event EThe complement of event E is the event E, such that:\[ \overline{E} = U - E \]Thus, E is the set of elements of U that do not belong to E.Impossible event\[ E = ∅ \]Ex.: Rolling the number 7 on a die.Certain event\[ E = U \]It is the sample space itself.Ex.: Getting heads or tails when tossing a coin.Mutually exclusive events\[ E ∩ E’= ∅ \]Getting heads and tails in a single coin toss.Probability that event E occursIt is the ratio between the number of elements in set E and the number of elements in the universal set.\[ P(E) = \frac{n(E)}{n(U)} = \frac{\text{number of elements in set E}}{\text{number of elements in set U}} \]This definition is only valid if all the elements of have the same probability of occurring, that is, if the sample space is equiprobable.Probability is a real number that ranges from 0 to 1, usually expressed as a percentage for easy visualization.ExamplesExample 1: What is the probability of obtaining the number 3 when rolling a die?Solution:\[ U = \{1,2,3,4,5,6\} \]\[ E = \{3\} \]Answer:\[ P(3) = \frac{n(E)}{n(U)} = \frac{1}{6} \]Example 2: : What is the probability of obtaining an even number when rolling a die?E = {2,4,6} ---------------→ n(E) = 3U = {1,2,3,4,5,6} -------→ n(U) = 6\[ P(even) = \frac{n(E)}{n(U)} = \frac{3}{6} = \frac{1}{2} \]Example 3: Probability of obtaining “heads” when tossing a coin?E = {heads}--------------------→ n(E) = 1U = {heads, tails} ----------→ n(U) = 2\[ P(heads) = \frac{n(E)}{n(U)} = \frac{1}{2} \]Example 4 : What is the probability of obtaining a sum of 5 when rolling two dice?E = { (1,4) , (2,3) , (3,2) , (4,1) } --------→ n(E) = 4U = 6x6=36 -------------------------------→ n(U) = 36\[ P(\text{sum 5}) = \frac{n(E)}{n(U)} = \frac{4}{36} = \frac{1}{9} \]Addition of probabilitiesLet A and B be two events of the universal set U that have an intersection. The probability that event A or event B occurs is given by:\[ P( A \cup B) = P(A) + P(B) - P(A \cap B) \]If events A and B do not have an intersection, the probability that event A or event B occurs is given by:\[ P( A \cup B) = P(A) + P(B) \]ExamplesExample 5: What is the probability of obtaining a 6 or an even number of points when rolling a die?\[ U: \{1,2,3,4,5,6\} \]\[ \text{Event A: {6} - probability that it will occur:} \frac{1}{6} \]\[ \text{Event B: {2,4,6}-probability that it will occur:} \frac{3}{6} \]\[ \text{A \cap B: {6} probability that it will occur:} \frac{1}{6} \]\[ P(A \cup B) = \frac{1}{6} + \frac{3}{6} - \frac{1}{6} = \frac{3}{6} = \frac{1}{2} \]Example 6: In a group of 36 fans, 18 support Flamengo, 10 support Vasco, 6 support Fluminense, and 2 support Botafogo. Choosing one of these fans at random, what is the probability that he supports Flamengo or Botafogo?\[ \text{Probability of supporting Flamengo:} P(F): \frac{18}{36} \]\[ \text{Probability of supporting Botafogo:} P(B): \frac{2}{36} \]\[ \text{Probability of supporting Flamengo or supporting Botafogo:} \frac{18}{36} + \frac{2}{36} = \frac{20}{36} =\frac{5}{9} \]Multiplication of probabilitiesWhen we have two independent events A and B, the probability that one of them and the other occur is given by the relation:\[ P(A \cap B) = P (A) * P(B) \]ExamplesExample 7: In two rolls of a die, what is the probability of obtaining an even number on the first roll and an odd number on the second roll?A: the result of the 1st roll is even and B: the result of the 2nd roll is odd. Noting that A and B are independent, because the information that A occurred does not change the probability that B occurs, we have:\[ P(A \cap B) = P (A) * P(B) \]\[ \frac{3}{6} · \frac{3}{6} \]\[ \frac{9}{36} = \frac{1}{4} \]Example 8: Making successive rolls of a die until obtaining 6 on a roll, what is the probability that three attempts will be necessary?For three attempts to be necessary, the 1st attempt must not obtain 6 points, the 2nd must not either, and the 3rd must. Since the result of each roll is independent of the results of the other rolls, the requested probability is:\[ P(\text{getting 6 on the 3rd attempt}) = \frac{5}{6} · \frac{1}{6} · \frac{5}{6} = \frac{25}{216} \]Example 11: What is the probability that, when rolling the die, we obtain face 4 twice in a row?\[ P(\text{face 4 twice in a row}) = \frac{1}{6} · \frac{1}{6} = \frac{1}{36} \]Example 12: In a room there are 20 girls: 3 of them are studious, 11 are from Rio de Janeiro, and 7 do not have a boyfriend. What is the probability that João chooses from among them one who is studious, from Rio de Janeiro, and does not have a boyfriend?\[ \text{prob. of being studious} = \frac{3}{20} \]\[ \text{prob. of being from Rio de Janeiro} = \frac{11}{20} \]\[ \text{prob. of not having a boyfriend} = \frac{7}{20} \]\[ \text{probability of being studious, from Rio de Janeiro, and not having a boyfriend} = \frac{3}{20} · \frac{11}{20} · \frac{7}{20} = \frac{231}{8000} \]Proposed ExercisesWhen rolling a die, what is the probability of obtaining:an even number of points?a number of points less than or equal to 4?What is the probability of tossing a coin and obtaining tails?What is the probability of rolling one white die and one blue die and getting:a sum of the points equal to 7?2 points on the blue die?What is the probability of tossing two coins and getting the same results?What is the probability of drawing a ball from an urn containing 5 red balls and 1 white ball and getting a red ball?A coin is biased in such a way that getting heads is twice as likely as getting tails. Calculate the probability of:getting heads when tossing this coin?getting tails when tossing this coin?A die is biased, so that the probability of observing any even number is the same, and the probability of observing any odd number is also the same. However, an even number is three times more likely to occur than an odd number. When rolling this die, what is the probability of:getting a prime number?getting a multiple of 3?getting a number less than or equal to 3?What is the probability that, from a 52-card deck, we draw a diamond or a king?A group of 100 university students is made up of 52 engineering students, 27 medical students, 10 philosophy students, and the rest law students. If one member of the group is chosen at random, what is the probability that they are an engineering or medical student?An urn contains 40 cards, numbered from 1 to 40. If we randomly draw one card from this urn, what is the probability that the number written on the card is a multiple of 4 or a multiple of 3?A die is biased, so that the probability of observing a number on the top face is proportional to that number. Calculate the probability of getting an even number or the number 6.In exercise 11, what is the probability of getting an even number or the number 5?ANSWER KEYa) 1/2 b) 2/31/2a) 1/6 b) 1/61/25/6a)2/3 b)1/3a)5/12 b)1/3 c)5/124/1379/1001/24/717/21
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