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 is all of the 52 choices.

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 falls on Monday.

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 falls on Thursday.

Which of the following are possible probabilities? % -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. ResponseFrequency Country656 Rap202 Top Jazz48 Classic Rock 1051

Problem: What is the probability that a randomly selected student will prefer Jazz? ResponseFrequency Country656 Rap202 Top Jazz48 Classic Rock prefer Jazz Total: 2567 Probability = 48/2567

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: What are the possible combinations? 1. Jaime and Armando 2. Jaime and Aaron 3. Jaime and Ulysses 4. Armando and Aaron 5. Armando and Ulysses 6. 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

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? Make charts to simplify:

Hope Sara Luis Ariel Elizabeth Sara Hope Luis Ariel Elizabeth Hope and Sara Hope and Luis Hope and Elizabeth Hope and Ariel Sara and Hope Sara and LuisSara and Elizabeth Sara and Ariel Luis and Sara Luis and HopeLuis and Elizabeth Luis and Ariel Ariel and Sara Ariel and HopeAriel and Elizabeth Ariel and Luis Elizabeth and Sara Elizabeth and HopeElizabeth and Ariel Elizabeth and Luis 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: HHHTTT HTTTHH HTHTHT HHTTTH How many possibilities include at least one tail? How many possibilities total? Probability: 7/8 7 8

Probabilities: OR What is the probability of drawing a king or a heart from a deck of 52 cards?

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

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? 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 Related or Unrelated? Related

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 Probability = 3/9 which reduces to 1/3 RGGRBR RBGBBG RRGGBB

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 The probability that a family does both is Find the probability that a family visits New York or rides the 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

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 = – 0.40 = 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/ sides

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/ sides