Probability

1. Probability

2. Randomness “Random” in statistics is a description of a kind of order that emerges only in the long run. A phenomenon is random if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large of repetitions.

3. Randomness Independenceis when the outcome of one trial must not influence the outcome of any other. The probabilityof an outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions.

4. Probability Models • Sample Space S: The set of all possible outcomes. • Event: the outcome or a set of outcomes or a random event. A subset of the sample space.

5. Relative Frequency • Probability of an event = Relative frequency • If f is the frequency of an event and n is the sample size then is the relative frequency • For example if we take a random sample of 100 students and we find that 30 of the students have an A in English class we could say that the relative frequency of students with A’s in English is

6. Law of Large Numbers • In the long run, as the sample size increases and increases, the relative frequency of outcomes get closer and closer to the theoretical (or actual) probability value. • Examples: Casinos Insurance

7. Equally likely outcomes • Probability of an event= Number of outcomes favorable to event Total number of outcomes • All the possible outcomes are called the Sample Space (S). Example: The probability of flipping a coin and getting heads is 1 2

8. Venn Diagrams S = sample space A = some event B = some event Visually represent probabilities using area

9. Probability Notation • P(A)=probability of event A • P(B)=probability of event B • P(A)=Number of outcomes favorable to A Total Number of Outcomes • P(B)=Number of outcomes favorable to B Total Number of Outcomes

10. Properties of Probability • The probability of an event is always between 0 and 1. • A probability of 0 can only occur if there is no chance for the favorable outcome. • Example the probability of rolling a 7 on a dice is 0 • A probability of 1 can only occur if every outcome is favorable. • Example: The probability of getting heads or tails on the flip of a coin is 1.

11. Multiplication Principle 1 2 3 4 5 6 Heads SAMPLE SPACE: H1,H2,H3,H4,H5,H6,T1,T2,T3,T4,T5,T6 1 2 3 4 5 6 Tails • What is the probability we get a heads and an even number? Suppose we flip a coin and toss a die. What is the sample space (S)? P(head and even) = 3/12 = .25

12. Multiplication Principle a ways to do one task b ways to do another task a*b ways to do both tasks.

13. Complement of Events • The sum of the probability of all outcomes in a sample space is 1 • The probability that an event A does not occur is called the complement and is written P(AC ) • P(AC )=1 – P(A) • For example when rolling a dice the probability of not rolling a 2 is

14. Probability Rules Rule 1. The probability P(A) of any event A satisfies 0 < P(A) < 1 Rule 2. If S is the sample space in a probability model, then P(S) = 1 Rule 3. The complement if any event A is the event A doesn’t occur, written as Ac. P(Ac)=1-P(A) Rule 4. Two events A and B are disjoint (mutually exclusive) if they have no outcomes in common and so can never occur simultaneously. P(A or B) = P(A) + P(B). { Addition rule for disjoint events}

15. Disjoint Events Suppose you are rolling a dice. What is the probability of getting a 2 or a 5? A = getting a 2 B = getting a 5 The events are disjoint – you can’t get a 2 and 5 together. P(A or B) = 1/6 + 1/6 = 2/6 = 1/3

16. Multiplication Rule • Events are independent if one event does not change the probability of the other event. • A = an event, B = an event. If A and B are independent then: P(A and B) = P(A) * P(B) or… Note: Disjoint events are NOT independent.

17. Multiplication Rule example • Suppose we roll a 12 sided die (numbered 1-12) and toss a coin (heads or tails). What is the probability we get a 10 and a heads? A = rolling a 10  P(A) = 1/12 B = heads  P(B) = 1/2 The events are independent so P(A and B) = (1/12) * (1/2) = 1/24

18. Replacement Standard Deck = 52 total cards 4 Queens in a deck. A = 1st choice is a Queen B = 2nd choice is a Queen Since we replace the 1st card we select, A and B are independent. Picking a Queen the 1st time has no effect on whether I get a Queen the next time. P(A and B) = P(A) * P(B) = 4/52 * 4/52 = .006 = .6% Suppose you pick a card from a standard deck of 52 cards. You look at the card and then place it back into the deck and select another card. What is the probability that both cards are Queens?

19. Mutually exclusive events • Mutually Exclusive Events (events cannot occur at the same time): P(A or B) = P(A) + P(B) P(A or B or C) = P(A) + P(B) + P(C) What if the events do occur at the same time?

20. Non-mutually exclusive events P(A or B) = P(A) + P(B) – P(A and B) or What is P(A or B) if A and B are not disjoint (mutually exclusive)?

21. Example No! First notice .6 + .5 > 1…both Mike and Tom can get As simultaneously. A = Mike gets an A B = Tom gets an A P(A or B) = P(A) + P(B) – P(A and B) = .6 + .5 - .3 = . 8 So there is a 80% chance that at least one gets an A. X = at least one gets an A P( Xc ) = 1-.8 = .2 • Mike estimates the probability he gets an A in AP Stats to be 0.6. Mike also estimates Tom’s probability of getting an A to be 0.5. Mike estimates the probability both get As to be 0.3. • Are the events of Mike getting an A or Tom getting an A disjoint? • What is the probability Mike or Tom get an A? • What is the probability neither gets an A?

22. Example Continued What is the probability Mike gets an A and Tom does NOT get an A? This is called a joint event.