Probability

Vocabulary



Probability



describes how likely it is that some event will happen

For example:



What is the probability of randomly picking the king of hearts from a deck of cards?

Vocabulary



Sample space: a combination of all of the possible choices

For example:



When drawing a card from a deck of 52 cards, there are 52 possible choices



The sample space 52 choices.

is all of the

Sample space

for drawing a card from a deck

Properties of Probabilities



The probability of any event must fall between 0 and 1.



0 < probability < 1

Properties of probabilities

 If something is

impossible

, then the probability is

0

.

 The probability that Thanksgiving will fall on a Monday is

0

.

 Thanksgiving

never

Monday.

falls on

Properties of probabilities

 If an event is

certain

, then the probability is

1

.

 The probability that Thanksgiving will fall on a Thursday is

1

.

 Thanksgiving

always

Thursday.

falls on

Which of the following are possible probabilities?

 0.2

 3.1

 3%  -2.3%  134%  1/3  3/2

Calculating Probabilities

 A pair of dice are rolled.

 What is the probability of rolling a seven?

 To calculate:  Number of ways a seven could be rolled Number of ways two dice can land

A pair of dice are rolled.

What is the probability of rolling a seven?

6/36 which reduces to 1/6

Situation:

 In a local survey, high school students were asked what kind of music they preferred. The results are printed.

Response Country Rap Top 100 Jazz Classic Rock Frequency 656 202 610 48 1051

Problem:

 What is the probability that a randomly selected student will prefer Jazz?

48 prefer Jazz Response Country Rap Top 100 Jazz Probability = 48/2567 Frequency 656 202 610 48 Classic Rock 1051

Total: 2567

Rap Concert:

 Cain has 3 tickets to a concert. Ulysses, Armando, Aaron, and Jaime would all like to go. Cain will randomly select two of them.

 What is the probability that Aaron and Jaime will get to go?

Rap concert:

 1.

2.

3.

4.

5.

6.

What are the possible combinations?

Jaime and Armando

Jaime and Aaron

Jaime and Ulysses Armando and Aaron Armando and Ulysses Ulysses and Aaron Probability that Jaime and Aaron will get to go = 1/6

Rap Concert:

 Cain has 3 tickets to a concert. Ulysses, Armando, Aaron, and Jaime would all like to go. Cain will randomly select two of them.

 What is the probability that Aaron or Jaime will get to go?

Rap concert:

       What are the possible combinations?

Jaime and Armando Jaime and Aaron Jaime and Ulysses Armando and Aaron Armando and Ulysses Ulysses and Aaron Probability that Jaime or Aaron will get to go = 5/6

Make charts to simplify:

 Two members from a five member committee are to be randomly selected to serve as chairman and secretary.  First person selected will be chairman. Second person selected will be secretary.

 Five members are: Hope, Sara, Luis, Elizabeth, and Ariel.

 What is the probability that Hope will be the chairman

and

Sara will be the secretary?

Hope Sara Luis Ariel Elizabeth Hope and Sara Hope and Luis Hope and Ariel Hope and Elizabeth Sara Luis Hope Luis Ariel Elizabeth Sara Hope Ariel Elizabeth Sara and Hope Sara and Luis Luis and Sara Luis and Hope Sara and Ariel Sara and Elizabeth Luis and Ariel Luis and Elizabeth Ariel Elizabeth Sara Hope Luis Elizabeth Sara Hope Luis Ariel Ariel and Sara Ariel and Hope Elizabeth and Sara Elizabeth and Hope Ariel and Luis Ariel and Elizabeth Elizabeth and Luis Elizabeth and Ariel

Probability = 1/20

Tossing three coins:

 When three coins are tossed, the probability of getting at least one tail is:  Start by finding the possible combinations: HHH HTT HTH HHT TTT THH THT TTH

How many possibilities include at least one tail?

7 How many possibilities total?

8 Probability: 7/8

Probabilities: OR

 What is the probability of drawing a king or heart from a deck of 52 cards?

a

OR means add (but do not count the same item twice)

 Probability of selecting a heart: 13/52  Probability of selecting a king: 4/52  Probability of selecting a heart or a king:  13/52 + 4/52 – 1/52 (because we can’t count the king of hearts twice) = 16/52

Probabilities: OR

 What is the probability of drawing a king or a heart from a deck of 52 cards?

AND means multiply: unrelated events

 When a coin is tossed and then a die is rolled, the probability of getting a tail on the coin and an odd number on the die is:  If the two events

do not effect one another

then find the probability of each separately and multiply

Unrelated events Tails Odd

Probability of getting tails: 1/2 Probability of odd number: 3/6

Unrelated events: Tail and Odd

 Probability of getting tails: 1/2  Probability of getting odd number: 3/6  Probability of getting tails AND odd number: =(1/2)(3/6) = 3/12 = 1/4

What is the probability of drawing a king AND then drawing a king again if the first card is replaced?

Related or Unrelated?

Unrelated

Probability of drawing the first king: 4/52 Probability of drawing the second king: 4/52 Answer : (4/52)(4/52) = 1/169

AND means multiply: related events

 What is the probability of drawing a king, not replacing it, and drawing a king again?:  If one event

effects the outcome of the other event

then find the probability of the first event, and then find the probability of the second event

remembering the effect of the first

What is the probability of drawing a king AND then drawing a king again if the first card is not replaced?

Related or Unrelated?

Related

Probability of drawing the first king: 4/52 Probability of drawing the second king: 3/51 (3 kings and 51 cards left) Answer : (4/52)(3/51) = 1/221

Sample Problems

Two dice are rolled. What is the probability of getting doubles or a sum of 10?

Probability of getting doubles or sum of 10:

 Probability of getting

doubles

: 6/36  Probability of getting

sum of 10

: 3/36  Number of rolls that are

both

doubles and sum of 10: (1 roll, the five and five roll)  Probability answer:   =

6

/36 +

3

/36 -

1

/36 = 8/36  = 2/9

 A bag contains a red bead, a green bead, and a blue bead. If a bead is selected and its color noted, and then it is replaced and another bead is selected, the probability that both beads will be of the same color is

R G R B RR G G R B GG B B R G BB

Probability = 3/9 which reduces to 1/3

 A box contains a penny, a nickel, a dime, and a quarter. If a coin is selected and then replaced and a 2 nd coin is selected, the probability of getting an amount greater than 11 cents is Probability = 10/16 which reduces to 5/8

 A box contains a penny, a nickel, a dime, and a quarter. If two coins are selected without replacement, the probability of getting an amount greater than 11 cents is Probability = 8/12 which reduces to 2/3

 The probability that a family visits New York is 0.64, and the probability that a family rides on the Subway is 0.50. The probability that a family does both is 0.40. Find the probability that a family visits New York or rides the subway.

NY SUBWAY BOTH 0.64

0.50

0.40

OR means add: probability of people who only went to NY + probability of people who only went to subway

NY 0.64

SUBWAY 0.50

BOTH 0.40

OR means add: probability of people who

only

went to NY + probability of people who

only

went to subway Remember: Some of the people who went to the NY also went to the subway, we don’t want to count those people twice. Therefore we can subtract the people who went to both.

Probability of going to NY OR subway = 0.64

+ 0.50

– 0.40

= 0.74

 If a pair of tetrahedral die are rolled, what is the probability that the sum will be 6?

How many sides on a tetrahedral dice?

Probability = 3/16 4 sides 1 1 1 1 3 1 2 4 2 2 2 2 3 4 1 2 3 3 3 3 3 4 1 2 4 4 4 1 4 2 4 3

 If a pair of tetrahedral die are rolled,

given that the sum is even

, what is the probability that the sum will be 6?

Probability = 3/8 4 sides 1 1 2 1 3 1 4 1 1 2 2 2 3 2 4 2 1 3 2 3 3 3 4 3 1 4 2 4 3 4 4 4