Binomial Distribution

And generaldiscrete probability distributions...

Random Variable

• A random variable assigns a number to a chance outcome or chance event.

• The definition of the random variable is denoted by uppercase letters at the end of alphabet, such as W, X, Y, Z.

• The possible values of the random variable are denoted by corresponding lowercase letters w, x, y, z.

Examples: Discrete random variables

• W = number of beers randomly selected student drank last night, w = 0, 1, 2, …

• X = number of aspirin randomly selected student took this morning, x = 0, 1, 2, ...

• Y = number of children in a family with children, y = 1, 2, 3, ...

Discrete Probability Distribution

• A discrete probability distribution specifies:– the possible values of the random variable, and– the probability that each outcome will occur

Example: Discrete probability distribution

• Let X = number of natural brothers PSU students have. X = 0, 1, 2, ...

• P(X=0) = 0.41

• P(X=1) = 0.45

• P(X=2) = 0.11

• P(X=3) = 0.03

Example: Graphically …

0 1 2 3

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

X, number of brothers

Pro

babi

lity

Example: Discrete probability distribution

• Let X = the number selected when a student picks a number between 0 and 9.

• If students pick number randomly, then the probability of picking any number is 0.10.

• That is, P(X = 0) = … = P(X = 9) = 0.10

Example: Binomial random variable

• Student told 3 statements -- 2 true, 1 false.

• Student tries to identify false statement.

• Student does this 3 different times.

• Let X = the number of times student correctly identifies the false statement. Then, X = 0, 1, 2 or 3.

• independent trials: a student’s success or failure on one try doesn’t affect success or failure on another try

Binomial Random VariableA special kind of discrete random variable having the following four characteristics:

• 2 possible outcomes denoted “success” or “failure”: student picks either false statement or true statement.

• p = P(“success”) is same for each trial: if just guessing, a student has probability of 1/3 of picking false statement.

• n identical “trials”: student tries to guess 3 times

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 Clinton should have been removed from office is 0.29.

Gallup poll surveys 960 Americans.

Let X = number of Americans in sample who think Clinton should have been removed from office.

Is X binomial?

Students pick one number between 0 and 9.

Let X = number of students who pick the number “7”

Example: Binomial r.v.

Outcome X P(X=x)

NNN 0 1(0.1)0(0.9)3

YNNNYNNNY

1 3(0.1)1(0.9)2

YYNYNYNYY

2 3(0.1)2(0.9)1

YYY 3 1(0.1)3(0.9)0

Let 3 students pick. Let Y = #7 and N = not #7

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

Guessing game. Let n = 3 and p = 0.33. Then:

P(X = x) = n!/[x!(n-x)!] × px × (1-p)n-x

P(X = 0) = 3!/[0!(3-0)!] × 0.330 × (0.67)3-0

= 6/(1×6) × 1 × 0.673 = 0.30

P(X = 1) = 3!/[1!(3-1)!] × 0.331 × (0.67)3-1

= 3 × 0.33 × 0.672 = 0.44

Example (continued)

P(X = 2) = 3!/[2!(3-2)!] × 0.332 × (0.67)3-2

= 3 × 0.1089 × 0.67 = 0.22

P(X = 3) = 3!/[3!(3-3)!] × 0.333 × (0.67)3-3

= 1 × 0.037 × 1 = 0.04

Note: 0.30 + 0.44 + 0.22 + 0.04 = 1

Using binomial probabilities to draw a conclusion

If students did just randomly guess which statement was false, we’d expect the random variable X to follow a binomial distribution?

Can we conclude that students did not guess the false statements randomly?

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.