• The 2010 American Community Survey estimates that 47.1% of women ages 15 and over are married.
• We can view this as a Bernoulli distribution with being married as our success criteria with a \(p = .471\)

• What is the probability that the third woman selected is the only one who is married.
• P(1st Success is the 3rd woman) = \((1-p)^2 * p\)

(1-.471)^2 * .471

## [1] 0.1318051

• The probability that the third woman is the only one married is 13.81%

• Assume that the women being married are independent events
• P(Three Married) = \(p^3\)

.471^3

## [1] 0.1044871

• There is a 10.44% that all three are married

• How many women would we expect to sample to find our first success?
• This is the mean of the Bernoulli Distribution: \(\mu = \frac{1}{p} = \frac{1}{.471}\)

1/.471

## [1] 2.123142

• We expect to sample 3 women to find one that is married. It's really (2.12) but how do you sample .12 women?

• To find the standard deviation we apply the formula \[ \sigma = \sqrt{\frac{1-p}{p^2}} \]
• Using our value for \(p = .471\) we get

sqrt((1-.471)/(.471^2))

## [1] 1.544212

• We get a standard deviation of 1.54

• This changes our \(p = .3\)
• How many would we have to sample to find a married women?

1/.3

## [1] 3.333333

• We would have to sample 4 (3.33 those darn fractional people) women to find one that is married.

• The new standard deviation is.

sqrt((1-.3)/(.3^2))

## [1] 2.788867

• The standard deviation is 2.79

Decreasing the probability of an event occurring serves to increase the mean of the distribution and it's standard deviation.