1. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events You have seen counts for RANDOM EVENTS fluctuate Geiger-Meuller counters clicking in response to a radioactive source the cosmic ray singles rate of one of your CROP detectors

2. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events Cosmic rays form a steady background impinging on the earth equally from all directions measured rates NOT literally CONSTANT long term averages are just reliably consistent These rates ARE measurably affected by Time of day Direction of sky Weather conditions

3. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events You set up an experiment to observe some phenomena …and run that experiment for some (long) fixed time… but observe nothing: You count ZERO events. What does that mean? If you observe 1 event in 1 hour of running Can you conclude the phenomena has a ~1/hour rate of occurring?

4. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events Random events arrive independently unaffected by previous occurrences unpredictably 0 sectime  A reading of 1 could result from the lucky capture of an exceeding rare event better represented by a much lower rate (~ 0? ). or the run period could have just missed an event (starting a moment too late or ending too soon).

5. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events A count of 1 could represent a real average as low as 0 or as much as 2 1 ± 1 A count of 2 2 ± 1? ± 2? A count of 37 37 ± at least a few? A count of 1000 1000 ± ?

6. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events This histogram displays, minute by minute, 2-fold coincidences between CROP detectors arranged in a 2-stack telescope.

7. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events Most readings are close to the 609.5 counts/minute average Note the zero is "suppressed“! (the vertical axis starts at 500, not 0) 500 These are actually minor fluctuations riding atop a nearly 600 tall base. 1 bin as low as 562 562 657 1 bin as high as 657

8. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events No sudden spikes up to 800; no drops down to 400. Is this good data? How do we decide if the fluctuations are reasonable or jumping too wildly?

9. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events DATETIMECounter1&2Counters2&3 11/17/200419:46:00660645 11/17/200419:47:00683622 11/17/200419:48:00656677 11/17/200419:49:00647658 11/17/200419:50:00650668 11/17/200419:51:00673620 11/17/200419:52:00659634 11/17/200419:53:00649647 11/17/200419:54:00642682 11/17/200419:55:00632642 11/17/200419:56:00688640 =AVERAGE(C2:C1145) =STDEV(

10. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events The standard deviation is a calculation of how far, on average, every data point is from the mean. If every reading were identical, the mean would be obvious and the standard deviation simply zero.

11. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events Height in inches of sample of 100 male adults 61 64 67 70 70 71 75 72 72 69 68 70 65 67 62 59 62 66 66 68 68 73 71 72 73 71 71 68 69 68 69 65 63 70 70 76 71 72 74 60 56 74 75 79 72 72 69 68 68 68 62 66 66 66 61 77 75 74 63 72 63 62 65 65 66 65 67 67 65 67 68 62 67 60 68 65 70 70 69 70 68 73 64 71 71 68 70 69 73 72 70 69 67 64 67 58 66 69 76 73 Frequency table of the distribution of heights 561 570 581 591 602 612 625 633 643 657 667 678 6812 698 7010 717 728 735 743 753 762 771 780 791

12. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events Range = x max -x min = 23 Mean = =67.20 Mode = 68 Median = 68.52

13. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events averages alone don’t show how tightly clumped together the data is The range can be misleading if the sample includes rare extreme data points.

14. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events

15. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events mean,  describe the spread in data by a calculation of the average distance each individual data point is from the overall mean  (x i –  ) 2 N-1  = i=1 N

16. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events Range = x max -x min = 23 Mean = =67.20  = 4.357

17. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events The added lines mark distances one and two standard deviations above and below the mean. For this data set Excel reported a STDEV of 20. So lines have been added at 609.5 ± 20.0 = 589.5 and 609.5 ± 40.0 = 569.5 629.5 649.5

18. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events A very small number (here 4) lie more than 2 STDEV away from the mean. Most fluctuations appear to lie within ±1 STDEV of the mean. A few data points fall between 1-2 STDEV. The STDEV describes to us how tightly clustered fluctuations are about the mean, and defines a limit to how widely they might range. Here we see none beyond 3 STDEV.

19. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events Vertical lines mark ranges ± 1, 2 STDEV from the mean. The rounded peak shows most of the data is within ±1 STDEV of the mean. Very rarely will counts be recorded >3 STDEV from the mean. Counts recorded in minute intervals, over a three hour period (same detectors, next day). Readings are bunched closely about the mean (615). Zero suppressed; little data below 500 (or above 700).

20. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events

21. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events In other words: 68% of the events fall within ±1 STDEV of the mean. 95% of the events fall within ±2 STDEV of the mean 99.7% of the events fall within ±3 STDEV of the mean Characteristic of this shape is that the region between µ  and µ+  contains 68% of the total area under the curve.

22. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events Random events occuring at a reproducible average rate are well understood statistically. These include coin flips dice rolls automobile accidents radioactive decays and cosmic rays. The probability of counting a number n occurrences of a purely random event (heads in repeated coin tosses, a particular face value of repeated rolls of a dice, or cosmic rays in 5 minute run) is given by the Poisson distribution

23. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events A “fair” coin is flipped at the start of a football game to determine which team receives the ball. The “probability” that the coin comes up HEAD s is expressed as A. 50/50“fifty-fifty” B. 1/2 “one-half” C. 1:1 “one-to-one” In betting parlance the odds are 1:1; we say the chances are 50/50, but the mathematical probability is ½.

24. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events There are 36 possible outcomes for the toss of 2 six-sided dice. If each is equally likely, the most probable total score of any single roll is? A. 4 B. 5 C. 6 D. 7 E. 8 F. 9

25. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events A green and red die are rolled together. What is the probability of scoring an 11? A. 1/4B. 1/6C. 1/8 D. 1/12E. 1/18F. 1/36

26. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events “Snake eyes” give the minimum roll. “Boxcars” give the maximum roll. The probability of rolling any even number, Probability(even) ______ Probability(odd), the probability of rolling any odd number. A. > B. =C. <

27. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events Die Total Number of ways to score die totals

28. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events A coin is tossed twice in succession. The probability of observing two heads (HH) is expressed as A. 1/2B. 1/4 C. 1D. 0

29. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events A coin is tossed twice in succession. The probability of observing two heads (HH) is expressed as A. 1/2B. 1/4 C. 1D. 0 It is equally likely to observe two heads (HH) as two tails (TT) T) True.F) False.

30. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events A coin is tossed twice in succession. The probability of observing two heads (HH) is expressed as A. 1/2B. 1/4 C. 1D. 0 It is equally likely to observe two heads (HH) as two tails (TT) T) True.F) False. It is equally likely for the two outcomes to be identical as to be different. T) True.F) False.

31. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events A coin is tossed twice in succession. The probability of observing two heads (HH) is expressed as A. 1/2B. 1/4 C. 1D. 0 It is equally likely to observe two heads (HH) as two tails (TT) T) True.F) False. It is equally likely for the two outcomes to be identical as to be different. T) True.F) False. The probability of at least one head is A. 1/2B. 1/4 C. 3/4D. 1/3

32. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events 2 3 4 5 6 7 8 9 10 11 12 654321654321 The probability of “throwing a 7” P (7) = 6/36 = 1/6 P (7±5) = ?

33. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events 2 3 4 5 6 7 8 9 10 11 12 654321654321 P (7) = 6/36 = 1/6 P (7±5) = 36/36 =1 P (7±1) =16/36 = 4/9 P (7±2) = 24/36 = 2/3 the full range!

34. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events If events occur randomly in time, the probability that the next event occurs in the very next second is as likely as it not occurring until 10 seconds from now. T) True. F) False.

35. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events P(1)  Probability of the first count occurring in in 1st second P(10)  Probability of the first count occurring in in 10th second i.e., it won’t happen until the 10th second ??? P(1) = P(10) ??? = P(100) ??? = P(1000) ??? = P(10000) ???

36. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events Imagine flipping a coin until you get a head. Is the probability of needing to flip just once the same as the probability of needing to flip 10 times? Probability of a head on your 1st try, P(1) = Probability of 1st head on your 2nd try, P(2) = Probability of 1st head on your 3rd try, P(3) = 1/2 1/4 1/8 Probability of 1st head on your 10th try, P(10) = 1/1024

37. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events A six-sided die is rolled repeatedly until it gives a 6. What is the probability that one roll is enough?

38. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events A six-sided die is rolled repeatedly until it gives a 6. What is the probability that one roll is enough? 1/6 What is the probability that it will take exactly 2 rolls?

39. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events A six-sided die is rolled repeatedly until it gives a 6. What is the probability that one roll is enough? 1/6 What is the probability that it will take exactly 2 rolls? (probability of miss,1st try)  (probability of hit)=

40. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events A six-sided die is rolled repeatedly until it gives a 6. What is the probability that one roll is enough? 1/6 What is the probability that it will take exactly 2 rolls? (probability of miss, 1st try)  (probability of hit)= What is the probability that exactly 3 rolls will be needed?

41. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events The probability of a single COSMIC RAY passing through a small area of a detector within a small interval of time  t can be very small: p << 1 cosmic rays arrive at a fairly stable, regular rate when averaged over long periods the rate is not constant nanosec by nanosec or even second by second this average, though, expresses the probability per unit time of a cosmic ray’s passage

42. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events would mean in 5 minutes we should expect to count about A.6,000 events B. 12,000 events C. 72,000 events D. 360,000 events E. 480,000 events F. 720,000 events 1200 Hz = 1200/sec Example: a measured rate of

43. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events would mean in 3 millisec we should expect to count about A.0 events B. 1 or 2 events C. 3 or 4 events D. about 10 events E. 100s of events F. 1,000s of events 1200 Hz = 1200/sec Example: a measured rate of 1 millisec = 10  3 second

44. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events would mean in 100 nanosec we should expect to count about A.0 events B. 1 or 2 events C. 3 or 4 events D. about 10 events E. 100s of events F. 1,000s of events 1200 Hz = 1200/sec Example: a measured rate of 1 nanosec = 10  9 second

45. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events The probability of a single COSMIC RAY passing through a small area of a detector within a small interval of time  t can be very small: p << 1 for example (even for a fairly large surface area) 72000/min=1200/sec =1200/1000 millisec =1.2/millisec = 0.0012/  sec =0.0000012/nsec

46. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events The probability of a single COSMIC RAY passing through a small area of a detector within a small interval of time  t can be very small: p << 1 The probability of NO cosmic rays passing through that area during that interval  t is A. p B. p 2 C. 2p D.( p  1) E. (  p)

47. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events The probability of a single COSMIC RAY passing through a small area of a detector within a small interval of time  t can be very small: p << 1 If the probability of one cosmic ray passing during a particular nanosec is P(1) = p << 1 the probability of 2 passing within the same nanosec must be A. p B. p 2 C. 2p D.( p  1) E. (  p)

48. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events The probability of a single COSMIC RAY passing through a small area of a detector within a small interval of time  t is p << 1 the probability that none pass in that period is ( 1  p )  1 While waiting N successive intervals (where the total time is t = N  t ) what is the probability that we observe exactly n events? × ( 1  p ) ??? ??? “misses” p n n “hits” × ( 1  p ) N-n N-n“misses”

49. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events While waiting N successive intervals (where the total time is t = N  t ) what is the probability that we observe exactly n events? P(n) = n C N p n ( 1  p ) N-n P(n) = e -Np

50. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events P(n) = e  4 If the average rate of some random event is p = 24/min = 24/60 sec = 0.4/sec what is the probability of recording n events in 10 seconds ? P(0) = P(4) = P(1) = P(5) = P(2) = P(6) = P(3) = P(7) = e -4 = 0.018315639 0.018315639 0.073262556 0.146525112 0.195366816 0.156293453 0.104195635 0.059540363

51. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events P(n) = e -Np Hey! What does Np represent?

52. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events  = Np P(n) = e  Poisson distribution probability of finding exactly n events within time t when the events occur randomly, but at an average rate of   (events per unit time)

53. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events  =1  =4  =8

54. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events Another abbreviation (notation): mean,  = x (the average x value) i.e.

55. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events Recall: The standard deviation  is a measure of the mean (or average) spread of data away from its own mean. It should provide an estimate of the error on such counts. or for short

56. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events The standard deviation  should provide an estimate of the error in such counts In other words  2 =   = 

57. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events Assuming any measurement N usually gives a result very close to the true  the best estimate of error for the reading is N We express that statistical error in our measurement as N ± N

58. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events Cosmic Ray Rate (Hz) Time of day 1000 500 0

59. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events How many pages of text are there in the new Harry Potter and the Half-Blood Prince? 562 What’s the error on that number? A. 0 B.  1 C.  2 D.  /562  23.7 E.  562/2 =  281

60. The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events A punted football has a hang time of 4.6 seconds. What is the error on that number? Scintillator is sanded/polished to a final thickness of 2.50 cm. What is the error on that number?