Probability Topics

1. Addition Rule2. Multiplication Rule3. Compliments4. Conditional Probability5. Permutation6. Combinations7. Expected value8. Geometric Probabilities9. Binomial Probabilities

Addition Rule for Non Mutually Exclusive Events

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(A or B) = P(A) + P(B) – P(A and B)

One card is drawn from a standard deck of cards. What is the probability that it is red or an ace?

= P(Red) + P(Ace) – P(Both Red and Ace)= 26/52 + 4/52 – 2/52 = 28/52

Multiplication Rule Finding the probability of more than one event.

The word “AND” is always used when describing the situation.

1) P(rolling a 4 and then a 2) = 1/6 *1/6 = 2.8%

2) P(rolling 3 odd #’s) = 3/6*3/6*3/6 = 12.5%

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Example 1 continued P(A1 AND A2) = P(A1)P(A2|A1)

P(A1) = 4/52

There are now 3 aces left in a 51-card pack

P(A2|A1) = 3/51

Overall: P(A1 AND A2) = (4/52) (3/51) = .0045

What’s the probability of pulling out two aces in a row from a deck of 52 cards?

If A is an event within the sample space S of an activity or experiment, the complement of A (denoted A') consists of all outcomes in S that are not in A.

The complement of A is everything else in the problem that is NOT in A.

Compliment:P(A') = 1 - P(A)

Conditional Probability

)()(

)|(BP

BAPBAP

and

measures the probability of an event given that another event has occurred

1% of the population has disease X.

If someone has the disease and gets tested the test is positive every time.

If a healthy person gets tested for disease X they will get a false positive 10% of the time.

If the lab comes back positive what will be the probability the person actually has the disease?

n = total number of itemsr = number chosen

)!(

!rn

nrpn

an arrangement of items in a particular order.

Permutations

Permutations Examples

A combination lock will open when the right choice of three numbers (from 1 to 30) is selected. How many different lock combinations are possible assuming no number is repeated?

2436028*29*30)!330(!30

330

27!30! p

Combinations

. 0 where nrrnr

nrCn

)!(!!

an arrangement of items in which order does not matter. There are always fewer combinations than permutations.

n = total number of itemsr = number chosen

Combinations ExampleTo play a particular card game, each

player is dealt five cards from a standard deck of 52 cards. How many different hands are possible?

960,598,21*2*3*4*548*49*50*51*52)!552(!5

!52552

5!47!52! C

a weighted average of all possible values where the weights are the probabilities of each outcome

1 1 2 2( ) ( ) ( ) ( ).n nE x x P x x P x x P x

Expected value

Example: expected valueprobability distribution of ER arrivals x is the number of arrivals in one hour

X 10 11 12 13 14P(x) .4 .2 .2 .1 .1

The average expected number each hour

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1

3.11)1(.14)1(.13)2(.12)2(.11)4(.10)(i

i xpx

Geometric Distribution want to find the number of trials for the 1st success

p = probability of successq = 1 – p = probability of failureX = # of trials until first success occurs

p(x) = qx-1p

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Two Ways to use the Geometric Model

#1: the probability of getting your first success on the x trail

p(x) = qx-1p#2: the number of trials until the first success is certain

p(x) =

The desired probability is: p(x) = qx-1p4 1p(4) (.75) (.25) 0.0117

EXAMPLE:

On Friday’s 25% of the customers at an ATM make deposits. What is the probability that it takes 4 customers at the ATM before the first one makes a deposit.

✔ Two Categories: Success: make a deposit

Failure: don’t make a deposit✔ Probability success same for each trial✔ Wish to find the probability of the first

n = number of trialsx = number of successesn – x = number of failuresp = probability of success in one trialq = 1 – p = probability of failure in one trial

BINOMIAL PROBABILITYfinding the probability of a specific

number of successes

EXAMPLE 2You are taking a 10 question multiple choice test. If each question has four choices and you guess on each question, what is the probability of getting exactly 7 questions correct?

• p = 0.25 = guessing the correct answer • q = 0.75 = guessing the wrong answer• n = 10• x = 7

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