Ten things about Probability AP Statistics, Second Semester Review

Random Variables A variable that takes on values that are unpredictable in the short term, but form a distribution in the long term. Random variable can be continuous or discrete

Continuous Random Variables: Density Curves Always non-negative Area between curve and x-axis equal to one. Proportion of area above a range of x-values is equal to the probability of the event happening Famous density curves: Normal distribution T-distribution Chi-square distribution

Density Curves Is this a valid density curve? P(0<x<.5)=?

Continuous Random Variables Normal Distributions follow the empirical rule: Approximately 68% of the distribution is found within 1 standard deviation of the mean Approximately 95% of the distribution is found within 2 standard deviations of the mean Approximately 99.7% of the distribution is found within 3 standard deviations of the mean

Normal Calculations P(Z<-1)=? P(Z>-1)=? P(0<Z<2)=? What Z value is associated with being at the 25th percentile?

Normal Calculations Let X=N(65.5,2.5) P(X<63)=? P(X>67)=? What value of X is associated with being a the 75th percentile?

Discrete Random Values: Probability Distributions P(X>3)=? µx=? σx=?

Binomial Distributions Binomial Situation: Binary, Independent, Number of trials (n), probability (p) of Success is constant (BINS) X=the number of successes X=B(n,p) Roll a die 10 times, count the number of sixes. X=B(10,1/6) Randomly select 100 people and count the number who are left-handed Y=B(100,.1)

Binary Distribution X=B(10,1/6) P(X=2)=? P(X<2)=? P(X>2)=?

Conditional Distributions P(pierced)= P(male)= P(pierced & male)= P(pierced|male)= P(male|pierced)= Are male and pierced independent events?

Marginal Distributions What is the distribution based on gender? What is the distribution of piercing?

Tree Diagrams (Independent Events) Rolling a “6” and then a “5”

Tree Diagrams (Non-independent Events) 4 Brown Socks, 6 Blue Socks

Combining Distributions

Two rounds of golf X=N(45,5), Y=N(40,10)

Sampling Distribution

5 Women Selected X=N(65.5,2.5)

Sampling Distribution

Survey of 500 likely voters p=.55