Transcript Counting Random Events, 1 - Cosmic Ray Observatory Project

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 HEADs
is expressed as
A. 50/50
B. 1/2
C. 1:1
The Cosmic Ray Observatory Project
“fifty-fifty”
“one-half”
“one-to-one”
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. 1/4
D. 1/12
The Cosmic Ray Observatory Project
B. 1/6
E. 1/18
C. 1/8
F. 1/36
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. 1/2
B. 1/4
C. 1
D. 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. 1/2
B. 1/4
C. 1
D. 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. 1/2
B. 1/4
C. 1
D. 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. 1/2
B. 1/4
C. 1
D. 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. 1/2
B. 1/4
C. 3/4
D. 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
61
68
56
63
68
64
73
74
62
73
67
71
75
65
64
70
72
79
65
71
70
73
72
66
71
71
71
72
65
68
75
71
69
67
70
72
68
68
67
69
72
69
68
65
73
69
68
68
67
72
68
69
62
68
70
70
65
66
62
69
65
63
66
67
67
67
70
66
60
64
62
70
61
68
67
59
76
77
65
58
62
71
75
70
66
66
72
74
70
69
66
74
63
69
76
68
60
72
70
73
Frequency table of the distribution of heights
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
1
0
1
1
2
2
5
3
3
7
7
8
12
8
10
7
8
5
3
3
2
1
0
1
The Cosmic Ray Observatory Project
High Energy Physics Group
The University of Nebraska-Lincoln
Counting Random Events
Number of classes
K = 1 + 3.3 log10N
= 1 + 3.3 log10100
= 1 + 3.3×2 = 7.6  8
Frequency table of the distribution of heights
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
The Cosmic Ray Observatory Project
1
0
1
1
2
2
5
3
3
7
7
8
12
8
10
7
8
5
3
3
2
1
0
1
2
5
11
22
30
20
8
2
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.
The Cosmic Ray Observatory Project
F) False.
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) =
???
=
???
=
???
=
???
The Cosmic Ray Observatory Project
P(10)
P(100)
P(1000)
P(10000)
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)=
 5  1  5
   
 6  6  36
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)=
 5  1  5
   
 6  6  36
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 Dt
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 = 0.0012/msec
=0.0000012/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 Dt
can be very small:
p << 1
The probability of NO cosmic rays passing
through that area during that interval Dt is
B. p2
A. p
D.( p - 1)
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C. 2p
E. ( 1 - p)
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 Dt
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
D.( p - 1)
The Cosmic Ray Observatory Project
B. p2
C. 2p
E. ( 1 - p)
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 Dt 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 = NDt )
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 Dt 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 = NDt )
what is the probability that we observe
exactly n events?
pn
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 Dt 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 = NDt )
what is the probability that we observe
exactly n events?
pn
n “hits”
The Cosmic Ray Observatory Project
× ( 1 - p )???
??? “misses”
High Energy Physics Group
The University of Nebraska-Lincoln