Warm-Up 1. Expand (x 2 – 4) 7 1. Find the 8 th term of (2x + 3) 10

Probability, Odds and The Law of Large Numbers

Basics of Probability Experiment – any observation of random phenomenon. Experiment – any observation of random phenomenon. Outcomes – the different possible results of the experiment. Outcomes – the different possible results of the experiment. Sample Space – The set of all possible outcomes for an experiment. Sample Space – The set of all possible outcomes for an experiment. Ex) What would be the sample space of 3 children being born to a family and we note the birth order with respect to sex.

Find the sample space We select 1 card from a standard deck of 52 and then without returning the card, we select a second card. We select 1 card from a standard deck of 52 and then without returning the card, we select a second card.

What is an event? An Event is a subset of the sample space. An Event is a subset of the sample space. The event of having 2 girls and 1 boy would be a subset of the sample space we found earlier. {bgg, gbg, ggb} The event of having 2 girls and 1 boy would be a subset of the sample space we found earlier. {bgg, gbg, ggb}

Empirical Probability The probability of an outcome in a sample space is a number between 0 and 1. The probability of an outcome in a sample space is a number between 0 and 1. P(E) = the sum of the probabilities of the outcomes that make up E. P(E) = the sum of the probabilities of the outcomes that make up E. To find the probability of an event ocurring P(E) = the # of times E occurs_______ To find the probability of an event ocurring P(E) = the # of times E occurs_______ the # of times the experiment is performed the # of times the experiment is performed This ratio is the relative frequency.

Empirical information from an experiment What is the probability that the patient will develop severe side effects? What is the probability that the patient will develop severe side effects? What is the probability that a patient receiving the flu vaccine will experience no side effects? What is the probability that a patient receiving the flu vaccine will experience no side effects? SideEffects # of Times None67 Mild25 Severe8

Theoretical information can also be used to determine probability 1. We flip 3 fair coins. What is the probability of each outcome in the sample space? 1. We draw a 5-card hand randomly from a standard 52-card deck. What is the probability that we draw one particular hand?

When each outcome in a sample space is equally likely to occur Then the probability of each outcome occurring is Then the probability of each outcome occurring is For an event E in the sample space: For an event E in the sample space:

What is the probability of: Rolling a total of 4 when rolling two fair dice? Rolling a total of 4 when rolling two fair dice? Using the FCP we know that there are 36 possible outcomes. Using the FCP we know that there are 36 possible outcomes. E = {(1,3), (2,2), (3,1)} E = {(1,3), (2,2), (3,1)}

What is the probability of: Drawing a 5-card hand and all 5 cards are hearts? Drawing a 5-card hand and all 5 cards are hearts? Flipping 3 coins and getting 2 heads and a tail in any order? Flipping 3 coins and getting 2 heads and a tail in any order?3/8

Probability & Genetics 2 parents are carriers of a disease (d). Let (n) represent the normal gene. Create a Punnett square to display the possible genes of a child. 2 parents are carriers of a disease (d). Let (n) represent the normal gene. Create a Punnett square to display the possible genes of a child. What is the probability that a child will have the disease? What is the probability that a child will have the disease? Parent n1 d P a n r nn nn nd nd e n d t2 dn dn dd dd

Odds Odds are another way of stating the likelihood of an event. Odds are another way of stating the likelihood of an event. Expressed as a ratio Expressed as a ratio Successful event: Not a successful event Successful event: Not a successful event Suppose 2 of 7 events are successful. Suppose 2 of 7 events are successful. Probability: 2/7 or 0.29 Probability: 2/7 or 0.29 Odds in favor: 2/5 or 2:5 Odds in favor: 2/5 or 2:5 Odd against: 5/2 or 5:2 Odd against: 5/2 or 5:2

Mutually Exclusive Events When two events are mutually exclusive, you can add to find the probability that either one occurs. When two events are mutually exclusive, you can add to find the probability that either one occurs. For mutually exclusive events A and B: For mutually exclusive events A and B: P(A or B) = P(A) + P(B) P(A U B) P(A U B) If not mutually exclusive then: If not mutually exclusive then: P(A U B) = P(A) + P(B) – P(A B)

Compound Event Event made up of two or more events that can happen at the same time or one after the other. Event made up of two or more events that can happen at the same time or one after the other. Events can be independent or dependent Events can be independent or dependent Independent events – one event does not affect another event (rolling a die and tossing a coin) P(A and B)=P(A)*P(B) Independent events – one event does not affect another event (rolling a die and tossing a coin) P(A and B)=P(A)*P(B) Dependent events – one event affects another event (drawing two cards without replacement) P(A and B)=P(A)*P(B after A) Dependent events – one event affects another event (drawing two cards without replacement) P(A and B)=P(A)*P(B after A)

Cards Draw one card from a standard deck Which example is MUTALLY EXCLUSIVE EVENTS? What is the probability that the card is an ACE or a JACK? P(ace or jack)=? What is the probability that the card is an ACE or a JACK? P(ace or jack)=? What is the probability that the card is a FACE card or a spade? P(face card or spade)=? What is the probability that the card is a FACE card or a spade? P(face card or spade)=?

Who will win the prize? A school has 45 K, 55 1 st graders, 60 2 nd graders and 55 3 rd graders. The school has 15 teachers and 5 administrators. A radio station is giving away prizes at the school. What is the probability that the winner is: P(1 st grader)=?P(3 rd grade)=? P(teacher)=?P(K or adm)=?

Suppose the events are NOT mutually exclusive P(jack or spade)? P(jack or spade)? P(freshman or girl)? P(freshman or girl)? P(even or less than 5)? P(even or less than 5)? P(red or heart)? Draw one card from a standard deck P(red or heart)? Draw one card from a standard deck

Are the events disjoint (mutually exclusive)? P(A or B)=P(A) + P(B) P(A or B)=P(A) + P(B) Even or Odd Even or Odd K and 3 rd grader K and 3 rd grader P(A or B)=P(A) + P(B) – P(A and B) P(A or B)=P(A) + P(B) – P(A and B) Red and heart Red and heart Face card and spade Face card and spade Freshman and girl Freshman and girl Administrator and Male Administrator and Male

Find the probability: P(jack or spade)? Draw one card from a standard deck P(jack or spade)? Draw one card from a standard deck P(freshman or girl)? 45 Freshmen, 30 girls and 15 boys P(freshman or girl)? 45 Freshmen, 30 girls and 15 boys P(even or less than 5)? Roll die. P(even or less than 5)? Roll die. P(red or heart)? Draw one card from a standard deck P(red or heart)? Draw one card from a standard deck

Complementary Events Two events are complementary if they are mutually exclusive and together they include all the possibilities. Two events are complementary if they are mutually exclusive and together they include all the possibilities. Heart and Not a Heart Heart and Not a Heart Rain and Not rain Rain and Not rain Six and Not a Six Six and Not a Six Example: Find P(not a face card) when a card is randomly chosen from a standard deck.

The Law of Large Numbers (LLN) The LLN says that in the long run relative frequency of repeated independent events gets closer and closer to the true relative frequency as the number of trials increases. The LLN says that in the long run relative frequency of repeated independent events gets closer and closer to the true relative frequency as the number of trials increases. Note: The is not a Law of Averages. A common (mis)understanding of the LLN is that random phenomena are suppose to compensate somehow for whatever has happened in the past. So, if results have fallen to one side of what is expected are future results “due” in order to average out? (NO) Note: The is not a Law of Averages. A common (mis)understanding of the LLN is that random phenomena are suppose to compensate somehow for whatever has happened in the past. So, if results have fallen to one side of what is expected are future results “due” in order to average out? (NO) For example: For example: A pregnant lady has five daughters. The probability that child #6 is a girls is STILL 0.50 The sex of the baby is not dependent on previous children. A pregnant lady has five daughters. The probability that child #6 is a girls is STILL 0.50 The sex of the baby is not dependent on previous children. You strike out 8 times in a row. Are you due a hit? The probability that you will next time at bat is still 0.50 You strike out 8 times in a row. Are you due a hit? The probability that you will next time at bat is still 0.50 Flip a coin 1 time and get 8 heads. The probability the next flip is a tail is still 0.50 The coin does not know what has happened in the past. Flip a coin 1 time and get 8 heads. The probability the next flip is a tail is still 0.50 The coin does not know what has happened in the past.

The LLN does… promise that given a very large number of trials (in the long run) the distribution of subsequent results will eventually overwhelm any recent drift away from what is expected. promise that given a very large number of trials (in the long run) the distribution of subsequent results will eventually overwhelm any recent drift away from what is expected. The long run is a long time The long run is a long time

LLN You flip a coin 5 times and get 5 heads. Suppose you continue to flip the coin 100 times and end up with 54 heads and 51 tails. The lesson of the LLN is that random processes do not need to compensate in the short run to get back to the right long- run probabilities. The lesson of the LLN is that random processes do not need to compensate in the short run to get back to the right long- run probabilities.

In the Long Run… If the probabilities do not change and the events are independent, the probability of the next trial is always the same, no matter what has happened up to then. If the probabilities do not change and the events are independent, the probability of the next trial is always the same, no matter what has happened up to then. There is no Law of Averages for the short term. There is no Law of Averages for the short term. Relative frequencies settle down in the long run and we then can officially give the name “probability” to that value. Relative frequencies settle down in the long run and we then can officially give the name “probability” to that value.

0<probability<1 Roll one die Roll one die P(odd)= P(odd)= P(5 or 6)= P(5 or 6)= P(less than five)= P(less than five)= P(not greater than one)= P(not greater than one)= Roll two dice Roll two dice P(3 and 5)= P(3 and 5)= P(sum less than 10)= P(sum less than 10)= P(two evens)= P(two evens)=

Vocabulary Important to learn the terms and definitions for this unit of study. Important to learn the terms and definitions for this unit of study. Read the book and study the examples. Read the book and study the examples. Check HW answers with the back of the book. Check HW answers with the back of the book.

Warm-Up 1. Courtney is in charge of purchasing the following items for Christmas Angels: 3 stuffed animals, 2 action figures, and 5 books. If she has a choice of 10 stuffed animals, 15 action figures, and 20 books, how many ways can she make her selections? 2. Given a 10-sided die (decahedron): a. What is the probability of rolling a number greater than 2? b. What are the odds of rolling a 7? 195,350,400 ways 4/5 1:9

Given two 11-sided dice: What is the probability of rolling a sum of 7? What is the probability of rolling a sum of 7? What is the probability of rolling a sum larger than 3? What is the probability of rolling a sum larger than 3? Given a standard deck of 52 cards: 5. What is the probability of drawing a diamond or a face card? 6. What is the probability of drawing 3 diamonds in a row? (no replacement) 6/ /121 11/26 11/850 0r 0.13