STAT200 Elementary Statistics for applications

Transcription

1 STAT200 Elementary Statistics for applications Lecture # 12 Dr. Ruben Zamar Winter 2011 /

2 Randomness Randomness is unpredictable

3 Randomness Randomness is unpredictable With some structure nonetheless

4 Randomness Randomness is unpredictable With some structure nonetheless Example: coin toss Each toss in unpredictable

5 Randomness Randomness is unpredictable With some structure nonetheless Example: coin toss Each toss in unpredictable The frequency of Heads and Tails in a very large number of tosses approaches ½ (empirically observed)

6 Probability For a certain event Probability = proportion of times the event would occur if we were to perform a very large number of independent repetitions

7 Example - discussion?

8 Clicker question The statement There is a 60% chance of rain in Vancouver this afternoon means that: (A) It will rain in 60% of the Vancouver region (B) Although it might not rain, we think it will (C) Out of many days with similar meteorological conditions as today, it rained in 60% of them (D) Out of many cities in the world, it will rain in 60% of them.

9 Probability Our concept of probability involves the concept of frequency This is called the Frequentist approach

10 Probability Our concept of probability involves the concept of frequency This is called the Frequentist approach There is also the Bayesian approach which is conceptually very different

11 DISCUSSION

12 Discussion

13 Clicker question This use of chance refers to (A) frequency in a large number of replications (B) the respondent's perception of the likelihood of such an outcome (C) proportion of people in the population that match the condition of interest (D) pure random luck

14 Probability model Sample Space: list of possible outcomes S = {0,1,2,3,4,5,6,7,8,9,10} Event: a subset of outcomes A = {At most 3} = {0,1,2,3} Probability: a measure of how likely is any given event.

15 Probability model Probability: a measure of how likely is any given event. A probability can be assigned to each outcome (finite / countable case) P(0) =?, P(1) =?,,P(10)=? P(x) = C(10,x) p^x (1-p)^(10-x)

16 Probability model A list of possible outcomes Heads, Tails Sample space set of all possible outcomes S = { Disease present, Disease absent } S = { Heads, Tails }

17 Probability model Sample space Toss a coin twice S = { (H,H), (H,T), (T, H), (T, T) } Record the number of students with a smart phone out of a random sample of 10 of them S = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }

18 Probability model Height (in cm) of a randomly chosen student at UBC S =? Cannot list all possible heights... S = [0.5, 250]? or S = [0, 300]? There are infinitely many possible outcomes

19 Probability model Event a collection of possible outcomes Example: Toss a coin twice Description A = { exactly one head is observed } = { (H, T), (T, H) } Explicit list of outcomes

20 Probability model Example: Count and record the number of students with a smart phone out of a random sample of 10 students A = { less than half of the polled students had a smart phone } A = {? }

21 Probability model Example: Count and record the number of students with a smart phone out of a random sample of 10 students A = { less than half of the polled students had a smart phone } A = { 0,1,2,3,4}

22 Properties of probabilities Should be numbers between 0 and 1

23 Properties of probabilities Should be numbers between 0 and 1 All possible outcomes should have a collective probability of 1

24 Properties of probabilities Should be numbers between 0 and 1 All possible outcomes should have a collective probability of 1 If A and B are two mutually exclusive events P( A U B ) = P( A ) + P( B ) P(A) = probability that the event A occurs

25 Properties of probabilities 0 P( A ) 1 for all events A P( S ) = 1, where S = sample space P( A U B ) = P( A ) + P( B ) whenever A B = Ø P( A c ) = 1 - P( A ) where A c is the complement of A

26 Example Toss a coin twice S = { (H,H), (H,T), (T, H), (T, T) }

27 Example Toss a coin twice S = { (H,H), (H,T), (T, H), (T, T) } A = { no tails } = {..?.. }

28 Example Toss a coin twice S = { (H,H), (H,T), (T, H), (T, T) } A = { no tails } = {..?.. } P( A ) =?

29 Example Toss a coin twice S = { (H,H), (H,T), (T, H), (T, T) } A = { no tails } = { (H,H), } P( A ) =?

30 Example Toss a coin twice S = { (H,H), (H,T), (T, H), (T, T) } A = { no tails } = {(H,H) } P( A ) = ¼ = 0.25

31 Probabilities in finite sample spaces When the sample space is finite, it is sufficient to assign a probability to each outcome S = { s 1, s 2, s 3,, s k } An event A is a set containing some (or all) of the outcomes.

32 Probabilities in finite sample spaces For example: A = {s 3, s 7, s 10, s 23 } P( A ) = P( s 3 ) + P( s 7 ) + P( s 10 ) + P( s 23 )

33 Example Record the number of students with a smart phone in a random sample of 10 students S = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } P( 0 ) =? P( 1 ) =? P( 2 ) =? P( 8 ) =? P( 9 ) =? P( 10 ) =? P(x) = C(10,x) p^x (1-p)^(10-x), p = 0.60