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”

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

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

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.

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 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

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 Frequency table of the distribution of heights

The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events Number of classes K = log 10 N = log = ×2 = 7.6  8 Frequency table of the distribution of heights

The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events If events (the emission of an  particle from a uranium sample, or the passage of a cosmic ray through a scintillator) occur randomly in time, repeated measurements of the time between successive events should follow a “normal” (Gaussian or “bell-shaped”) curve T) True. F) False.

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.

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) ???

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) =

The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events Probability of a head on your 1st try, P(1) =1/2 Probability of 1st head on your 2nd try, P(2) =1/4 Probability of 1st head on your 3rd try, P(3) =1/8 Probability of 1st head on your 10th try, P(10) =

The Cosmic Ray Observatory Project High Energy Physics Group The University of Nebraska-Lincoln Counting Random Events What is the total probability of ALL OCCURRENCES? P(1) + P(2) + P(3) + P(4) + P(5) + =1/2+ 1/4 + 1/8 + 1/16 + 1/32 +

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?

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?

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)=

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?

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 for example (even for a fairly large surface area) 72000/min=1200/sec =1.2/millisec = /  sec = /nsec

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)

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)

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?

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? p n n “hits”

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? p n × ( 1  p ) ??? n “hits” ??? “misses”