AP Test Review #3

Focus Binomial and Geometric Distributions Basic Probability Tree Diagrams

Bernoulli Trial

A distribution where: The outcomes are success or failure Each trial is independent The probability of success is constant

Binomial and Geometric Distributions

Binomial and geometric distributions are Bernoulli trials

Binomial Dist. – the probability of a certain number of successes out of a set number of trials (n,p,k)

Geometric Dist. – the probability that the 1st success occurs on the kth try. (p,k)

Binomial Formulas

Probability of k success out of n trials (pdf)

Probability of k or fewer successes out of n trials – use binomial cdf!

Mean:

Standard Deviation:

knk ppk

n

)1(

npq

np

Geometric Formulas

Probability of 1st success on the kth trial

Cdf = probability of 1st success occurs on or before the kth trial

Mean:

pp k 11

p

1

Example 1:Assume the probability of getting a blue token

on each turn in a certain game is constant and equal to 0.35. Find:

a) The probability of getting 4 blue tokens in 7 turns.

b) The probability of getting at least 4 blue tokens in 7 turns.

c) Mean and standard deviation for the number of blue tokens in 11 turns.

d) The probability the 1st blue token occurs on the 5th turn.

e) The probability that it takes less than 4 turns to get the 1st blue token

f) The mean number of turns to get the 1st blue token.

Answers

a) P = .14424b) P = .19985c = 3.85 and = 1.58193d) P = .06248e) P = .7254f = 2.857

Basic Probability formulas

)(1)(

)(

)(

)()()()(

APAP

AP

BApABP

BAPBPAPBAP

c

Union (or)

Conditional (limit the total)

Complement

Example 2The probability of going to the junior Prom is

47% and the probability of going to the senior Ball is 39%. In addition, the probability of going to both the Junior Prom and Senior Ball is 18%. Find

a) P(junior prom or senior ball)b) P(junior prom|senior ball)c) P(not going junior prom nor senior ball)d) P(going to senior ball but not junior prom)e) P(senior ball|junior ball)

Answers

a) .47 + .39 - .18 = .69b) .18/.39 = .462c) 1 - .69d) .39 - .18 = .21e) .18/.47 = .383

Last Example

The probability that a Freshman entering college goes straight to a 4-year school is 38%. The probability that a student that goes straight to a 4-year school graduating in 4 years is 27%. A Freshman entering college that goes to a 2-year school to start and graduating in 4 years is 19%.

a) Find the probability of graduating in 4 years.

b) Find the probability of a student starting at a 2-year college and graduating in 4 years.