Download Binomial distribution

Binomial distribution
And general probability
distributions...
Random variables
• A number is assigned to an outcome that is
determined by chance.
• Definition denoted by uppercase letters at
the end of alphabet, such as X, Y, Z
• Possible values denoted by corresponding
lowercase letters x, y, z
Examples: random variables
• X = number of beers randomly selected
student drank last night, x = 0, 1, 2, …
• Y = weight of a randomly selected Big Mac,
0.20 x 0.30 pound
• Y = time spent daily on phone of randomly
selected student, 0 x 1440 minutes
Discrete
probability distribution
All possible outcomes of the r.v. and the
probability that each outcome will occur.
Example
Toss a fair die. Let X = value on die.
Then P(X=x) = 1/6, x =1, 2, 3, …, 6
Continuous
probability distribution
All possible outcomes of the r.v. and a formula
that allows us to determine the probability of a
range of values.
Example
Let X = blood alcohol level of randomly
selected student.
Use formula to find P(X  0.10)
Example: binomial r.v.
• Pick one number between 0 and 9.
• Let X = number of students picking certain
number.
• If random picking, then probability random
student picks special number is 0.10.
Binomial r.v.
A special kind of discrete r.v having the following
characteristics:
• n identical “trials”: n students pick # between 0 and 9
• 2 possible outcomes denoted “success” or “failure”:
student picks special # or not
• independent trials: what one student picks doesn’t affect
what another student picks
• p = P(“success”) is same for each trial: if random picking,
each student has probability of 0.10 of picking special #
Is X binomial?
Probability student smokes pot regularly is
0.25.
College administrator surveys students until
finds one who smokes pot.
Let X = number of students surveyed
Is X binomial?
Unknown to quality control inspector, crate
of 50 light bulbs contain 3 defective bulbs.
QC inspector randomly selects 5 bulbs
(“without replacement”).
Let X = number of defective bulbs in
inspector’s sample.
Is X binomial?
Unknown to us, the probability an American
thinks House Judiciary Committee is
mishandling the “Clinton thing” is 0.78.
Gallup poll surveys 960 Americans.
Let X = number of Americans in sample
who think HJC is mishandling the Clinton
thing.
Recall example
• Pick one number between 0 and 9.
• Let X = number of students picking certain
number.
• If random picking, then probability random
student picks special number is 0.10.
Example: binomial r.v.
Let 3 students pick. Y = special #, N = not special #
Outcome
X
P(X=x)
NNN
0
1(0.1)0(0.9)3
YNN
NYN
NNY
YYN
YNY
NYY
YYY
1
3(0.1)1(0.9)2
2
3(0.1)2(0.9)1
3
1(0.1)3(0.9)0
Binomial probability distribution
P(X = x)
= (# of ways x occurs) × px × (1-p)n-x
= n!/[x!(n-x)!] × px × (1-p)n-x
Where “n-factorial” is defined as
n!= n (n-1) (n-2) … 1
and
0! = 1
Examples: n!
5! = 5 × 4 × 3 × 2 × 1 = 120
4! = 4 × 3 × 2 × 1 = 24
3! = 3 × 2 × 1 = 6
2! = 2 × 1 = 2
1! = 1
Example: binomial formula
P(X = x) = n!/[x!(n-x)!] × px × (1-p)n-x
Assume n = 3 and p = 0.2. Then:
P(X = 0) = 3!/[0!(3-0)!] × 0.20 × (0.8)3-0
= 6/(1×6) × 1 × 0.83 = 0.512
P(X = 1) = 3!/[1!(3-1)!] × 0.21 × (0.8)3-1
= 3 × 0.2 × 0.82 = 0.384
Example (continued)
P(X = 2) = 3!/[2!(3-2)!] × 0.22 × (0.8)3-2
= 3 × 0.04 × 0.8 = 0.096
P(X = 3) = 3!/[3!(3-3)!] × 0.23 × (0.8)3-3
= 1 × 0.008 × 1 = 0.008
Note: 0.512 + 0.384 + 0.096 + 0.008 = 1 
Using binomial probabilities to
draw a conclusion
If students do indeed pick a number
between 0 and 9 randomly, how likely is it
that we would observe the sample we did?
Can we conclude that students do not pick
numbers randomly?
Using binomial probabilities to
draw a conclusion
Could the space shuttle Challenger disaster of
January 28, 1986 have been better predicted?
And therefore prevented?
Moral
• Probability calculations are used daily to
draw conclusions and make important
decisions.
• Calculated probabilities are accurate only if
the assumptions made are indeed correct.
• Always check to see if your assumptions
are reasonable.