Probability

The probability of an event is a measure of the likelihood of its occurrence.

An event is an outcome from an experiment.

A probability model lists the different outcomes from an experiment and their corresponding probabilities.

To construct probability models, we need to know the sample space of the experiment. This is the set S that lists all the possible outcomes of the experiment.

Determine the sample space resulting from the experiment of rolling a die.

S = {1, 2, 3, 4, 5, 6}

Properties of Probabilities

For a sample space S e e en 1 2, ,

1 0 1. ( ) for all events P e ei i

2 1 21

. ( ) ( ) ( ) ( )

= 1

P e P e P e P ei ni

n

Determine which of the following are probability models from rolling a single die.

Outcome Probability1 0.32 0.13 0.054 0.25 0.156 0.25

Not a probability model. The sum of all probabilities is not 1.

Outcome Probability1 0.22 0.23 0.24 0.25 0.26 0

All probabilities between 0 and 1 inclusive and the sum of all probabilities is 1.

Outcome Probability1 0.252 0.13 0.354 0.155 0.26 -0.05

Not a probability model. The event “roll a 6” has a negative probability.

Theorem Probability for Equally Likely Outcomes

If an experiment has n equally likely outcomes, and if the number of ways an event E can occur is m, then the probability of E is

P EE m

n( ) Number of ways that can occur

Total number of possible outcomes

A classroom contains 20 students: 7 Freshman, 5 Sophomores, 6 Juniors, and 2 Seniors. A student is selected at random. Construct a probability model for this experiment.

P F( ) 720

P Soph( ) 520

P Jr( ) 620

P Sr( ) 220

Theorem Additive RuleFor any two events and

E F

P E F P E P F P E F( ) ( ) ( ) ( )

if and are mutually exclusive.

P E F P E P F

E F

( ) ( ) ( )

What is the probability of selecting an Ace or Diamond from a standard deck of cards?

P P( (Ace) =4

52 Diamond) =

1352

1

1314

P

P

(Ace or Diamond)

= (Ace) + P(Diamond) - P(Ace and Diamond)

113

14

152

1652

413

Let S denote the sample space of an experiment and let E denote an event. The complement of E, denoted E, is the set of all outcomes in the sample space S that are not outcomes in the event E.

Theorem Computing Probabilities of Complementary Events

If E represents any event and E represents the complement of E, then

P E P E( ) ( ) 1

The probability of having 4 boys in a four child family is 0.0625. What is the probability of having at least one girl?

Sample Space: {4 boys; 3 boys, 1 girl, 2 boys, 2 girls; 1 boy, 3 girls; 4 girls}

E = “at least one girl”

E = “4 boys”

P(E) = 1 - P(E) = 1 - 0.0625 = 0.9375

What is the probability of obtaining 3 of a kind when 5 cards are drawn from a standard 52-card deck?

P(3 of a kind) CC

( , )( , )

4 3 48 4452 5

0 00325 0 325%. .This answer from the text slides is just wrong. For correct values to this and similar questions see either of the poker sites:

http://www.math.sfu.ca/~alspach/comp18/

http://www.pvv.ntnu.no/~nsaa/poker.html