4.2
Probability Models
probability model
 a description of a random phenomenon in the language of
mathematics
the description of a random variable has two components:
a list of possible outcomes
a probability for each outcome
The sample space S
of a random phenomenon is the set of all possible outcomes.
random phenomenon→ toss a coin
S= {H, T}
random phenomenon→ let your pencil fall blindly into table of random digits
S= {0, 1, 2, 3, 4, 5 ,6, 7, 8, 9}
random phenomenon→ toss a coin four times and record the results
vague what constitutes an outcome?
random phenomenon→ toss a coin four times and record the results of each of 4 tosses in
order
S= {HHHH, HHHT, HHTH, HTHH, THHH…}
16 possibilities
random phenomenon→ toss a coin four times and count the number of heads
S= {0, 1, 2, 3, 4}
random phenomenon→ computer generates random number between 0 and 1
S= {all numbers between 0 and 1}
event
 an outcome or a set of outcomes of a random phenomenon (an event is a subset of
the sample space)
random phenomenon→ toss a coin four times and record the results of each of 4 tosses in
order
event A= “get exactly 2 heads”
event A expressed as a set of outcomes:
A= {HHTT, HTHT, HTTH, THHT, THTH, TTHH}
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Probability Rules
1) Any probability is a number between 0 and 1.
The probability P(A) of any event A satisfies 0
≤
P(A)
≤
1.
2) All possible outcomes together must have probability 1.
If S is the sample space in a probability model, then P(S) = 1.
3) If two events have no outcomes in common, the probability that one or the other occurs is the
sum of their individual probabilities.
Two events A and B are disjoint
if they have no outcomes in common and so can never occur
together.
If A and B are disjoint,
P(A or B) = P(A) + P(B)
This is the addition rule for disjoint events
.
EX)
random phenomenon toss a coin once (Aheads, B tails)
A and B disjoint no outcomes in common (can’t get heads and tails at same time)
P (H or T) =
P (H) + P (T)
0.5 + 0.5 = 1
EX)
random phenomenon toss a die (A1, B odd #)
A and B not disjoint have outcome in common (roll a 1)
P (A or B)
≠
P(A) + P(B)
4) The probability that an event does not occur is 1 minus the probability that the event does
occur.
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 Spring '11
 Bill
 Probability, Probability theory, random phenomenon toss

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