Lecture notes for probability. Math 124

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1 Lecture notes for probability Math 124

2 What is probability? Probabilities are ratios, expressed as fractions, decimals, or percents, determined by considering results or outcomes of experiments whose result is not deterministic. An experiment is an activity whose results can be observed and recorded. Each of the possible results of an experiment is an outcome. A set of all possible outcomes for an experiment is a sample space. The outcomes in the sample space cannot overlap. Any subset of a sample space is an event. We typically find probabilities, or likelihood, of events.

3 Examples A possible experiment is tossing a coin. The outcomes for this experiment are: heads (H) and tails (T). The sample space for the experiment will consist of these two outcomes: S = {H, T }. Another possible experiment is tossing a die. The sample space for this experiment is {1, 2, 3, 4, 5, 6}. A possible event would be tossing an even number, A = {2, 4, 6}. Another possible event is tossing a number bigger than 4, B = {5, 6}.

4 What is experimental probability?

5 What is experimental probability? The experimental probability of an event is the relative frequency with which it occurs in an experiment.

6 What is experimental probability? The experimental probability of an event is the relative frequency with which it occurs in an experiment. For example, if you toss a coin 50 times and heads comes up 22 times, the experimental probability of tossing heads is 22/50 or 44%.

7 What is theoretical probability?

8 What is theoretical probability? While experimental probability shows what actually happened, theoretical probability predicts what should happen.

9 What is theoretical probability? While experimental probability shows what actually happened, theoretical probability predicts what should happen. For an experiment with non-empty sample space S with equally likely outcomes, the probability of an event A is given by P(A) = Number of elements of A Number of elements of S = n(a) n(s). Note that this only works if the outcomes are equally likely.

10 What is theoretical probability? While experimental probability shows what actually happened, theoretical probability predicts what should happen. For an experiment with non-empty sample space S with equally likely outcomes, the probability of an event A is given by P(A) = Number of elements of A Number of elements of S = n(a) n(s). Note that this only works if the outcomes are equally likely. There are many situations in which it is impossible to find theoretical probability. For example, to find the likelihood that you will be hit by lightning, you would have to know how frequently people are hit by lightning. On the other hand, the probability of winning the lottery is purely theoretical.

11 What is the experimental probability of getting a sum of seven, based on your recorded data from the game?

12 What is the experimental probability of getting a sum of seven, based on your recorded data from the game? Answers will vary. I rolled some virtual dice, and I got the sum of seven, 7/30 of the time.

13 What is the theoretical probability of getting a sum of seven?

14 What is the theoretical probability of getting a sum of seven? It may appear that all sums are equally likely. This is not the case. There are six ways to get a sum of 7, so the theoretical probability is 6/36=1/6.

15 For some event, when both its experimental probability and theoretical probability are possible to determine, will the two probabilities be equal?

16 For some event, when both its experimental probability and theoretical probability are possible to determine, will the two probabilities be equal? Probabilities only suggest what will happen in the long run. This concept is called The Law of Large Numbers or sometimes Bernoulli s Theorem: If an experiment is repeated a large number of times, the experimental probability of a particular outcome approaches its theoretical probability as the number of repetitions increases.

17 For some event, when both its experimental probability and theoretical probability are possible to determine, will the two probabilities be equal? Probabilities only suggest what will happen in the long run. This concept is called The Law of Large Numbers or sometimes Bernoulli s Theorem: If an experiment is repeated a large number of times, the experimental probability of a particular outcome approaches its theoretical probability as the number of repetitions increases. That means that maybe you got a sum of 7 more often than any other sum in your game, but maybe you did not. But if we all played the game for three hours and put all our data together, we could be almost certain that the experimental probability of rolling a 7 would be close to 1/6.

18 Gambling The Law of Large Numbers works in favor of casinos. Though one or two patrons may have a running streak, in the long run the casino always wins, because all games have a higher probability of losing than winning.

19 Why is a probability limited to numbers between, and including, 0 and 1? When is it 0? When is it 1?

20 Why is a probability limited to numbers between, and including, 0 and 1? When is it 0? When is it 1? An event that has no outcomes is an impossible event and has probability 0. If event B is the empty set, then it has no elements and its probability is the ratio P(B) = 0 n(s). For example, the event of rolling a sum larger than 12 with two dice has probability 0.

21 Why is a probability limited to numbers between, and including, 0 and 1? When is it 0? When is it 1? An event that has no outcomes is an impossible event and has probability 0. If event B is the empty set, then it has no elements and its probability is the ratio P(B) = 0 n(s). For example, the event of rolling a sum larger than 12 with two dice has probability 0. An event that has probability 1 is a certain event. If A is a subset of the sample space S, the greatest number of elements that A can have is the number of elements in the sample space, S. So the probability of A in this case would be P(A) = n(a) n(s) = n(s) n(s) = 1.

22 Why is a probability limited to numbers between, and including, 0 and 1? When is it 0? When is it 1? An event that has no outcomes is an impossible event and has probability 0. If event B is the empty set, then it has no elements and its probability is the ratio P(B) = 0 n(s). For example, the event of rolling a sum larger than 12 with two dice has probability 0. An event that has probability 1 is a certain event. If A is a subset of the sample space S, the greatest number of elements that A can have is the number of elements in the sample space, S. So the probability of A in this case would be P(A) = n(a) n(s) = n(s) n(s) = 1. For any set A that is a proper subset of S, n(a) is greater than or equal to 0 but less than n(s), so the probability of A is between 0 and 1.

23 Suppose you calculate the probability of each possible outcome of an experiment, and add these probabilities. What should the sum be?

24 Suppose you calculate the probability of each possible outcome of an experiment, and add these probabilities. What should the sum be? The probability of an event is equal to the sum of the probabilities of disjoint events whose union is the event.

25 Suppose you calculate the probability of each possible outcome of an experiment, and add these probabilities. What should the sum be? The probability of an event is equal to the sum of the probabilities of disjoint events whose union is the event. For example, the probabilities of rolling 1, 2, 3, 4, 5, or 6 with a die are all equal to 1/6. When you add them up, you get 1.

26 Suppose the probability of an event A is 2/3. What is the probability of event A not occurring?

27 Suppose the probability of an event A is 2/3. What is the probability of event A not occurring? Two events are mutually exclusive if they can t both happen at the same time. For example, you can roll an odd and even number at the same time.

28 Suppose the probability of an event A is 2/3. What is the probability of event A not occurring? Two events are mutually exclusive if they can t both happen at the same time. For example, you can roll an odd and even number at the same time. Two mutually exclusive events whose union is the sample space are complementary events. If A is an event, the complement of A, written A (or A C ), is also an event. If A is an event and A is its complement, then P( A) = 1 P(A).

29 Suppose the probability of an event A is 2/3. What is the probability of event A not occurring? Two events are mutually exclusive if they can t both happen at the same time. For example, you can roll an odd and even number at the same time. Two mutually exclusive events whose union is the sample space are complementary events. If A is an event, the complement of A, written A (or A C ), is also an event. If A is an event and A is its complement, then P( A) = 1 P(A). The probability of A not occurring in the example above is 1/3.

30 So what about the two-dice game? There is no one best strategy, especially since theoretical probability does not guarantee that what we predict to happen will actually happen. However, it is wisest to put more counters on the more likely numbers, for example, three counters on 7, two counters on 6 and 8, and 1 counter each on 4, 5, 9, and 10.

The enumeration of all possible outcomes of an experiment is called the sample space, denoted S. E.g.: S={head, tail}

The enumeration of all possible outcomes of an experiment is called the sample space, denoted S. E.g.: S={head, tail} Random Experiment In random experiments, the result is unpredictable, unknown prior to its conduct, and can be one of several choices. Examples: The Experiment of tossing a coin (head, tail) The Experiment