Answer.
Here
p
=
1
6
, so
np
= 30 and
p
np
(1

p
) = 5. Then
P
(
S
n
>
50)
≈
P
(
Z >
4)
,
which is very small.
Example.
Suppose a drug is supposed to be 75% eﬀective. It is tested on 100 people. What is the probability
more than 70 people will be helped?
Answer.
Here
S
n
is the number of successes,
n
= 100, and
p
=
.
75. We have
P
(
S
n
≥
70) =
P
((
S
n

75)
/
p
300
/
16
≥ 
1
.
154)
≈
P
(
Z
≥ 
1
.
154)
≈
.
87
.
(The last ﬁgure came from a table.)
When
b

a
is small, there is a correction that makes things more accurate, namely replace
a
by
a

1
2
and
b
by
b
+
1
2
. This correction never hurts and is sometime necessary. For example, in tossing a coin 100
times, there ispositive probability that there are exactly 50 heads, while without the correction, the answer
given by the normal approximation would be 0.
Example.
We toss a coin 100 times. What is the probability of getting 49, 50, or 51 heads?
Answer.
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 Spring '09
 wen
 Normal Distribution, Probability, Probability theory, Chisquare distribution, Gamma function

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