Probability, Odds and the Counting Principle

Created by BKHall 

Upgrade to
remove ads

Basic probability formulas and definitions

Odds against event E

P (fail) ÷ P (success)
Alternate notation P(fail) : P (success)

Expected value

P₁A₁ + P₂A₂ + ... + PnAn for events 1, 2, ... n
where Pn is the probability that n occurs and An is the value or cost if it does occur

Sample space

List of all possible outcomes of an event
Tree diagrams are helpful in determining sample space

Counting principle

The probability of two experiments are multiplied to determine the overall probability. Ex: a coin toss (2 outcomes) followed by the roll of a die (6 outcomes) has 2×6=12 outcomes (H1,H2,H3,H4,H5,H6,T1,T2,T3,T4,T5,T6)

Independent events

Occurrence of either event does not affect probability of the other
Ex: rolling a die or tossing a coin multiple times (with replacement)

Dependent events

Occurrence of one event depends on outcome of the other
Ex: drawing repeatedly from a deck of cards (without replacement)

Addition formula ("OR" problems)

P (A or B) = P (A) + P (B) - P (A and B)
Easy to see this with Venn diagram; subtracting the 3rd term prevents counting the "middles" twice

Permutation

Any ordered arrangement of a set of objects where order matters
Ex: abc, acb, bac, bca, cab, cba are all the permutaions of a,b,c

Number of permutations of n distinct items

n!

Number of permutations of n objects if some are identical

n! / (n₁! n₂! ... nr!) where n₁, n₂... are groupings of identical items
Ex: bag of 9 balls w/ 3 red, 3 yellow, and 3 blue has 9! / 3! 3! 3!

Combination

An arrangement of a set of objects where order does not matter
Ex: abc and cba are not distinct but the same combination of a,b,c

Number of permutations for selecting r items from among n

nPr = n! / (n-r)!
Ex: {a,b,c} select 2 from 3 → 3! / (3-2)! = 6 (ab,ac,ba,bc,ca,cb)

Number of combinations for selecting r items from among n

nCr = n! / r! (n-r)!
Ex: {a,b,c} select 2 from 3→ 3! / 2! (3-2)! = 3 (ab,ac,bc)

Multiplication of probabilities ("AND" problems)

P (A and B) = P(A) × P(B) , assuming event A has occurred
Alternate notation: P (B | A), probability of B given A

Find P(B | A)

A has already occurred, so that is the new sample space (denominator) in a probability calculation

Card dealing - Ex: probability of getting exactly three 6's in a five card hand

ways of choosing three 6's * ways of choosing two non-6's ÷ ways of choosing 5 from 52 = ₄C₃ * ₄₈C₂ / ₅₂C₅

Please allow access to your computer’s microphone to use Voice Recording.

Having trouble? Click here for help.

We can’t access your microphone!

Click the icon above to update your browser permissions above and try again

Example:

Reload the page to try again!

Reload

Press Cmd-0 to reset your zoom

Press Ctrl-0 to reset your zoom

It looks like your browser might be zoomed in or out. Your browser needs to be zoomed to a normal size to record audio.

Please upgrade Flash or install Chrome
to use Voice Recording.

For more help, see our troubleshooting page.

Your microphone is muted

For help fixing this issue, see this FAQ.

Star this term

You can study starred terms together

NEW! Voice Recording

Create Set