1MA01: Probability. Sinéad Ryan. November 12, 2013 TCD


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1 1MA01: Probability Sinéad Ryan TCD November 12, 2013
2 Definitions and Notation EVENT: a set possible outcomes of an experiment. Eg flipping a coin is the experiment, landing on heads is the event If an experiment is repeated many times then number of times event A happens number of times experiment repeated is called the Probability of event A, written P(A)
3 An example: Roll a die 1000 times and record the frequency with which each number appears giving something like VALUE FREQ P(1) = = P(2) = = =. Note that 6 P(i) = 1. i=1 Note that if P is a probability, 0 P 1
4 Theorem: The Principle of Equally Likely Outcomes If an experiment is conducted with n possible outcomes, all equally likely, and A is an event with k possible outcomes then P(A) = k n = #favourable outcomes for A #possible outcomes Example Consider a jar containing 7 black balls, 6 yellow balls, 4 green and 3 red balls. The jar is shaken and 1 ball removed without looking. What is the probability that the ball is red?
5 Theorem: The Principle of Equally Likely Outcomes If an experiment is conducted with n possible outcomes, all equally likely, and A is an event with k possible outcomes then P(A) = k n = #favourable outcomes for A #possible outcomes Example Consider a jar containing 7 black balls, 6 yellow balls, 4 green and 3 red balls. The jar is shaken and 1 ball removed without looking. What is the probability that the ball is red? P(red) = 3 20
6 Theorem: The Principle of Equally Likely Outcomes If an experiment is conducted with n possible outcomes, all equally likely, and A is an event with k possible outcomes then P(A) = k n = #favourable outcomes for A #possible outcomes Example Consider a jar containing 7 black balls, 6 yellow balls, 4 green and 3 red balls. The jar is shaken and 1 ball removed without looking. What is the probability that the ball is red? P(red) = 3 20 the ball is white?
7 Theorem: The Principle of Equally Likely Outcomes If an experiment is conducted with n possible outcomes, all equally likely, and A is an event with k possible outcomes then P(A) = k n = #favourable outcomes for A #possible outcomes Example Consider a jar containing 7 black balls, 6 yellow balls, 4 green and 3 red balls. The jar is shaken and 1 ball removed without looking. What is the probability that the ball is red? P(red) = 3 20 the ball is white? P(white) = 0 20 = 0
8 Theorem: The Principle of Equally Likely Outcomes If an experiment is conducted with n possible outcomes, all equally likely, and A is an event with k possible outcomes then P(A) = k n = #favourable outcomes for A #possible outcomes Example Consider a jar containing 7 black balls, 6 yellow balls, 4 green and 3 red balls. The jar is shaken and 1 ball removed without looking. What is the probability that the ball is red? P(red) = 3 20 the ball is white? P(white) = 0 20 = 0 the ball is either black, yellow, green or red?
9 Theorem: The Principle of Equally Likely Outcomes If an experiment is conducted with n possible outcomes, all equally likely, and A is an event with k possible outcomes then P(A) = k n = #favourable outcomes for A #possible outcomes Example Consider a jar containing 7 black balls, 6 yellow balls, 4 green and 3 red balls. The jar is shaken and 1 ball removed without looking. What is the probability that the ball is red? P(red) = 3 20 the ball is white? P(white) = 0 20 = 0 the ball is either black, yellow, green or red? P(b, y, g, r) = = 1
10 Addition, Complement and Multiplication Addition: suppose A and B are disjoint events. Then P(A or B) = P(A) + P(B) Two events are disjoint of they cannot occur in the same experiment. Eg. draw one card from a pack  drawing an ace and a king are disjoint events in this experiment since they cannot both occur. Let s look at an example of adding probabilities. Consider a deck of cards. Shuffle and draw 1 card. What is the probability of drawing an ace?
11 Addition, Complement and Multiplication Addition: suppose A and B are disjoint events. Then P(A or B) = P(A) + P(B) Two events are disjoint of they cannot occur in the same experiment. Eg. draw one card from a pack  drawing an ace and a king are disjoint events in this experiment since they cannot both occur. Let s look at an example of adding probabilities. Consider a deck of cards. Shuffle and draw 1 card. What is the probability of drawing an ace? P(ace) = 4 52 = 1 13
12 Addition, Complement and Multiplication Addition: suppose A and B are disjoint events. Then P(A or B) = P(A) + P(B) Two events are disjoint of they cannot occur in the same experiment. Eg. draw one card from a pack  drawing an ace and a king are disjoint events in this experiment since they cannot both occur. Let s look at an example of adding probabilities. Consider a deck of cards. Shuffle and draw 1 card. What is the probability of drawing an ace? P(ace) = 4 52 = 1 13 drawing a king?
13 Addition, Complement and Multiplication Addition: suppose A and B are disjoint events. Then P(A or B) = P(A) + P(B) Two events are disjoint of they cannot occur in the same experiment. Eg. draw one card from a pack  drawing an ace and a king are disjoint events in this experiment since they cannot both occur. Let s look at an example of adding probabilities. Consider a deck of cards. Shuffle and draw 1 card. What is the probability of drawing an ace? P(ace) = 4 52 = 1 13 drawing a king? P(king) = 4 52 = 1 13
14 Addition, Complement and Multiplication Addition: suppose A and B are disjoint events. Then P(A or B) = P(A) + P(B) Two events are disjoint of they cannot occur in the same experiment. Eg. draw one card from a pack  drawing an ace and a king are disjoint events in this experiment since they cannot both occur. Let s look at an example of adding probabilities. Consider a deck of cards. Shuffle and draw 1 card. What is the probability of drawing an ace? P(ace) = 4 52 = 1 13 drawing a king? P(king) = 4 52 = 1 13 not drawing a king?
15 Addition, Complement and Multiplication Addition: suppose A and B are disjoint events. Then P(A or B) = P(A) + P(B) Two events are disjoint of they cannot occur in the same experiment. Eg. draw one card from a pack  drawing an ace and a king are disjoint events in this experiment since they cannot both occur. Let s look at an example of adding probabilities. Consider a deck of cards. Shuffle and draw 1 card. What is the probability of drawing an ace? P(ace) = 4 52 = 1 13 drawing a king? P(king) = 4 52 = 1 13 not drawing a king? P(not a king) = = 12 13
16 Addition, Complement and Multiplication Addition: suppose A and B are disjoint events. Then P(A or B) = P(A) + P(B) Two events are disjoint of they cannot occur in the same experiment. Eg. draw one card from a pack  drawing an ace and a king are disjoint events in this experiment since they cannot both occur. Let s look at an example of adding probabilities. Consider a deck of cards. Shuffle and draw 1 card. What is the probability of drawing an ace? P(ace) = 4 52 = 1 13 drawing a king? P(king) = 4 52 = 1 13 not drawing a king? P(not a king) = = drawing an ace or a king?
17 Addition, Complement and Multiplication Addition: suppose A and B are disjoint events. Then P(A or B) = P(A) + P(B) Two events are disjoint of they cannot occur in the same experiment. Eg. draw one card from a pack  drawing an ace and a king are disjoint events in this experiment since they cannot both occur. Let s look at an example of adding probabilities. Consider a deck of cards. Shuffle and draw 1 card. What is the probability of drawing an ace? P(ace) = 4 52 = 1 13 drawing a king? P(king) = 4 52 = 1 13 not drawing a king? P(not a king) = = drawing an ace or a king? P(ace or king) = P(ace) + P(king) = 2 13
18 The assumption of disjoint events is crucial for addition of probabilities. So, eg. drawing an ace and drawing a heart are not disjoint so addition rules don t hold ie. P(ace or heart) = P(ace) + P(heart) Note above that P(king) = 1/ 13 and P(not king) = 12/ 13. These disjoint events are complements and since exactly one of these must happen then their probabilities sum to 1 P(king) + P(not king) = 1 P(not king) = 1 P(king) In general then, if A is an event then P(not A) = 1 P(A).
19 Multiplication rule Let A and B be 2 events, then P(A and B) = P(A)P(B A) P(B A) is called a conditional probability: it is the probability of event B occuring given event A has already occurred. Eg. Consider a deck of cards. Draw 2 cards. What is the probability the 1st card is a heart?
20 Multiplication rule Let A and B be 2 events, then P(A and B) = P(A)P(B A) P(B A) is called a conditional probability: it is the probability of event B occuring given event A has already occurred. Eg. Consider a deck of cards. Draw 2 cards. What is the probability the 1st card is a heart? P(A) = = 1 4
21 Multiplication rule Let A and B be 2 events, then P(A and B) = P(A)P(B A) P(B A) is called a conditional probability: it is the probability of event B occuring given event A has already occurred. Eg. Consider a deck of cards. Draw 2 cards. What is the probability the 1st card is a heart? P(A) = = 1 4 Given it s a heart what is the probability the 2nd card is a heart?
22 Multiplication rule Let A and B be 2 events, then P(A and B) = P(A)P(B A) P(B A) is called a conditional probability: it is the probability of event B occuring given event A has already occurred. Eg. Consider a deck of cards. Draw 2 cards. What is the probability the 1st card is a heart? P(A) = = 1 4 Given it s a heart what is the probability the 2nd card is a heart?p(b A) = = 4 17
23 Multiplication rule Let A and B be 2 events, then P(A and B) = P(A)P(B A) P(B A) is called a conditional probability: it is the probability of event B occuring given event A has already occurred. Eg. Consider a deck of cards. Draw 2 cards. What is the probability the 1st card is a heart? P(A) = = 1 4 Given it s a heart what is the probability the 2nd card is a heart?p(b A) = = 4 17 What is the probability both cards are hearts?
24 Multiplication rule Let A and B be 2 events, then P(A and B) = P(A)P(B A) P(B A) is called a conditional probability: it is the probability of event B occuring given event A has already occurred. Eg. Consider a deck of cards. Draw 2 cards. What is the probability the 1st card is a heart? P(A) = = 1 4 Given it s a heart what is the probability the 2nd card is a heart?p(b A) = = 4 17 What is the probability both cards are hearts? P(A and B) = P(A)P(B A) = 1 17
25 In the last example you could also use the P.E.L.O. and list all 2652 ways the 1st and 2nd cards could be dealt. Clearly using the multiplication rule is easier. Independence Events A and B are independent if P(B A) = P(B) and in this case P(A and B) = P(A)P(B). Note that independence is not the same thing as disjoint. A simple example of independence: roll a die twice. What is the probability of rolling 2 sixes? P(six and six) = P(A)P(B A) = P(A)P(B) = = 1 36 since the result of each roll is independent of the other. 1
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