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Binomial Probability Presentation
X=# of (successes)
n= ____________
Π= (decimals)
P(
) = P (X symbol #) = Bpd or Bcd (X, n, Π) = 0.xxxx
(words)
(<,>,≤, ≥,=)
(4decimals)
•
•
•
•
•
If (=) then Bpd
If (>) then Bcd, but for example (X>5 means 1-X≤5). This
is when they ask a more than type question.
If (<) then Bcd, but for example (X<5 means X≤4). This is
when they ask a less than type question.
If (≤) then Bcd. This is when they ask an at most type
question.
If (≥) then Bcd, but for example (X≥5 means 1-X≤4). This
is when they ask an at least type question.
The normal probability distribution has several
important properties:
1. The shape is determines by mean, μ and
standard deviation, σ
2. The highest point on the normal curve is
located at the mean, which is also the median
and the mode of the distribution.
3. The normal distribution is symmetrical: the
curve’s shape to the left of the mean is the
mirror image of its shape to the right of the
mean.
4. The tails of the normal curve extend to infinity
in both directions and never touch the
horizontal axis. However, the tails get close
enough to the horizontal axis quickly enough to
ensure that the total area under the normal
curve equals 1.
5. Since the normal curve is symmetrical, the area
under the normal curve to the right of the mean
(μ) equals the area under the normal curve to
the left of the mean, and each of these areas
equals 0.5.
Poisson Probability Presentation
X=# of (successes) IN (time of space)
λ= (given rate)
P(
) = P (X symbol #) = Ppd or Pcd (X, λ) = 0.xxxx
(words)
(<,>,≤, ≥,=)
(4decimals)
•
•
•
•
•
If (=) then Ppd
If (>) then Pcd, but for example (X>5 means 1-X≤5). This
is when they ask a more than type question.
If (<) then Pcd, but for example (X<5 means X≤4). This is
when they ask a less than type question.
If (≤) then Pcd. This is when they ask an at most type
question.
If (≥) then Pcd, but for example (X≥5 means 1-X≤4). This
is when they ask an at least type question.
Another case is from ___ to ___. This is
expressed for example P(2≤X≤5). Which means
probability is calculated P(X≤5) – P(X≤1).
Upper boundary = μ + 6σ
X= probability boundary
μ = Mean
σ = Standard Deviation
Lower boundary = μ - 6σ
P(
) = P (X symbol #) = Ncd (Lower, Upper,σ,μ) = 0.xxxx
(words)
(<,>,≤, ≥,=)
(4decimals)
E.g. What is the probability that the firm’s sales will be
less than $3.0 million?
$3.0 mil = upper boundary
μ - 6σ = lower boundary then use a value less than that
P(X<3)
E.g. What is the probability that the firm will have
sufficient sales to cover fixed costs (FC=$1.8mil)?
$1.8 mil = lower boundary
μ + 6σ = upper boundary then use a value greater than that
P(X≥1.8)
E.g. What is the probability that the firm’s sales will be
within $150,000 of the expected, i.e., mean, sales?
μ + 0.15= upper boundary
μ - 0.15 = lower boundary
P(2.35 ≤ X ≤ 2.65)
Prepared By: Sulosan Thangarajah
Suppose that a random variable x is
normally distributed with mean μ
and standard deviation σ. If a and b
are numbers on the real line, we
consider the probability that x will
attain a value between a and b.
That is we consider P(a ≤ x ≤ b)
which equals the area under the
normal curve with mean μ and
standard deviation σ corresponding
to the interval [a,b].