Bernoulli Trials

• Two Possible Outcomes– Success, with probability p– Failure, with probability q = 1 p

• Trials are independent.

Binomial Distribution• For n Bernoulli trials, the number of

successes X is a binomial random variable. The probability of k successes is given by the binomial probability formula:

• As k varies with fixed n and p, the binomial probabilities define a binomial probability distribution over {0, 1, 2, …, n}.

knk ppk

nkXP

1

Sampling Distribution of the Count in a SRS

• When the population is much larger than the sample, the count of X successes in a SRS of size n has approximately the Binomial(n,p) distribution (given that the true proportion of successes in the population is p).

• As a rule of thumb, we use the binomial sampling distribution for counts when the population is at least 10 times as large as the sample.

Mean and Standard Deviation of a Binomial RV X(i.e., of a sample count)

npX pnpX 1

Mean and Standard Deviation of a sample proportion, p̂

pp ˆ n

ppp

1ˆ

Law of Large Numbers

• Informal: If n is large, the proportion of successes in n Bernoulli trials will be very close to p.

• Formal: For Bernoulli trials with n and p, as n ,

for all > 0, where is the sample proportion.

1ˆ ppP

p̂

Binomial (n=100, p=1/2) Distribution

Number of Successes

30 40 50 60 70

Pro

babi

lity

0.00

0.02

0.04

0.06

0.08

0.10

Binomial (n=100, p=1/2) DistributionWith Normal Approximation Curve

Number of Successes

30 40 50 60 70

Pro

babi

lity

0.00

0.02

0.04

0.06

0.08

0.10

Normal Approximation• Draw a SRS of size n from a large population having proportion p

of successes. Let X be the count of successes in the sample and = X/n the sample proportion. When n is large, the sampling distributions of the two statistics are approximately normal:

• As a rule of thumb, we use the approximation for values of n and p such that np 10 and n(1p) 10.

p̂

n

pppNp

pnpnpNX

1, normalely approximat is ˆ

1, normalely approximat is

Example 1 – Normal Approximation of Counts

• Suppose you flip a balanced coin 1000 times. What is the probability of getting between 480 and 532 heads?

Distribution of Number of Heads in 1000 Flips of a Balanced Coin

Number of Heads

440 460 480 500 520 540 560

Pro

babi

lity

0.000

0.005

0.010

0.015

0.020

0.025

0.030

Distribution of Number of Heads in 1000 Flips of a Balanced Coin

Number of Heads

440 460 480 500 520 540 560

Pro

babi

lity

0.000

0.005

0.010

0.015

0.020

0.025

0.030

Distribution of Number of Heads in 1000 Flips of a Balanced Coin

Number of Heads

440 460 480 500 520 540 560

Pro

babi

lity

0.000

0.005

0.010

0.015

0.020

0.025

0.030

Approximate Normal Curve

Distribution of Number of Heads in 1000 Flips of a Balanced Coin

Number of Heads

477 478 479 480 481 482 483 530 531 532 533 534 535

Pro

babi

lity

0.000

0.005

0.010

0.015

0.020

0.025

0.030

Approximate Normal Curve

Example 2 – Normal Approximation of Proportions

• A corporation receives 100 applications for a position from recent college graduates in business. Assuming that these applications constitute a random sample of graduates in business, what is the probability that between 25% and 35% of the applicants are women if 30% of all recent college graduates in business are women?