Probability

Do Now

• What does probability mean to you and give an example.

• List all the possible outcomes of flipping a coin.

• List all the possible outcomes of rolling one die

• Next is the Monty Hall problem and Fraction, Percent, Decimal, Probability Chart.

Monty Hall Problem

• Movie 21• http://www.youtube.com/watch?v=cXqDIFUB7YU

• Explanation• http://www.youtube.com/watch?v=mhlc7peGlGg&feature=related

a. 1/1, 1.00

b. 9/10, .9

c. 999/1000, .999

d. 9/100, .09

e. 9/10000, .0009

f. 0/1, 0.00

a. 85%, 17/20

b. 67.68%, 423/625

c. 100%, 1/1

d. 0%, 0/1

e. .5%, 1/200

f. 10%, 1/10

g. 1%, 1/100

h. 400%, 4/1

a. .2, 20%

b. .375, 37.5%

c. .01, 1%

d. 1.00, 100%

e. 0, 0%

f. .368, 36.8%

4.

5.

6.

1

2

3

70%

.266

3/8

Lesson

• Probability is the study of random events. An understanding of this theory is essential to appreciating weather reports, the state lotteries, etc.

• In the Due Now we flipped a coin and listed all the possible results, which were heads and tails. If I flip a coin right now can you say for certain what the outcome will be?

No, You Can Not

• Why?

• You can’t say for certain what it is. This is called randomness. Meaning there is an uncertainty in any prediction. Probability gives us the means of measuring that uncertainty. Probability is based on outcomes, which is assigned a number between 0 and 1.

• Why is it assigned a number between one and zero?

Let us think about that.• Let’s looks at a fraction. A pie is split up in 3 equal slices. So 1/3, 1/3,

and 1/3. Added together equal 1 and when changed into a percent is 100%. If all 3 equal slices are eaten what do you have left? Zero, or 0% percent. Now depending on what question I ask or outcome I am looking for will determine if I get a number between 0 and 1 or 0% and 100%. If something happens its 100% and if it doesn’t happen its 0%.

1/3 1/3

1/3

cheese cheese

pepperoni

2 types of probabilities

• There are two types of probabilities Theoretical probabilities and Experimental probabilities.

• Any Idea what theoretical or experimental probabilities are? • Look at the words theoretical and experimental!

Theoretical Probability of Flipping a coin• If I flip a coin and want to see the theoretical probability of getting heads. I

would write as:

• The Probability of Heads = (the outcome I desire) (all the possible outcomes)

There is 1 desired outcome And 2 total possible outcomes.

So, it is 1/2. This is formally written as P(Heads) = 1 2

Experimental Probability

• Everyone take a penny and flip a coin. Record 20 times how many heads and how many tails.

Compare theoretical data to experimental data. • Due to randomness we don’t get our ideal result.

Under the Law of Large Numbers it says as the number of trials increases the experimental probability approaches the theoretical probability. Therefore the more times we flip that coin the closer our probability will approach 1/2 or .5 or 50%.

Can anyone guess the relationship between theoretical and experimental probabilities?

• With More TRIALS OURExperimental Probability The Theoretical Probability Test it!

Think about it!

• What if I asked what is the theoretical probability of getting tails?

• What about if I flip a coin what is the probability of getting heads or tails?

• What about if I flipped a coin what is the probability of getting neither heads nor tails?

Worksheet rolling two fair dice

• If we roll a pair of standard fair dice, what are all the possibilities? Fill in the chart. Now what is the probability of getting a sum of 5?

1,41,31,2 1,5 1,62,42,32,2 2,5 2,63,43,33,2 3,5 3,64,44,34,2 4,5 4,65,45,35,2 5,5 5,66,46,36,2 6,5

2,13,14,15,16,1

4/36 = 1/9 = .111111 or 11.1111%

Think about it, Make a Connection

6*6*6 = 216

• How many possibilities are there with 3 coins?

2*2*2 = 8

• So, 1 has 6 total outcomes 2 have 36 total outcomes ( 6 each ).

How many possible outcomes will 3 dice have?

Apply it

• For 3 dice there are 216 total possibilities, what is the probability of having the 3 dice sum to 5? (List them)

• (1, 2, 2), (1, 3, 1) The possibilities with a 1 on the first die.

• (2, 1, 2), (2, 2, 1) The possibilities with a 2 on the first die.• (3, 1, 1) The only possibility with a 3 on the first die.

• There are 6 desirable outcomes, so the probability of three dice summing to 5 is 6 out of 216 or 6/216, or 1/36.

(1, 1, 3)

Let’s switch it to a fair deck of cards. • How many total possible outcomes are there in a standard fair deck of

cards?• 52 possible outcomes/cards. • There are four different suits: clubs , diamonds , hearts , and spades . ♣ ♦ ♥ ♠

Each suit contains 13 cards labeled ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, and king. Two suits are red: diamonds and hearts. The other two suits are black: clubs and spades.

Example

• If you want to pick a heart, what is the desired outcome and the total number of possibilities?

You’re interested in 13 outcomes (hearts) out of the 52 possible outcomes. So the probability that your card is a heart is 13/52, or 1/4. (1/4 of the cards are hearts♥

SpinnerDiceCardGumball Worksheet

P(spin an odd number) = 5/10 = ½ = .5 = 50%

P(spin a number less than 4) = 6/10 = 3/5 = .6 = 60%

P(spin a number that is a perfect square) = 4/10 = 2/5 = .4 = 40%

P(dice sum to more than 8) = 10/36 = 5/18 = .2777778

7 & P(most likely sum): 6/36 = 1/6 = .1666667

P(rolling an even sum) = 18/36 = ½ = .5

P(draw a red card) = 26/52 = ½ = .5 = 50%

P(draw a 5 or a King) = 8/52 = 2/13 = .154 = 15.4%

P(draw a black 7) = 2/52 = 1/ 26 = .038 = 3.8%

P(blue gumball) = 155/600 = 31/120 = .258

P(blue or red gumball) = 275/600 = 11/24 = .458

P(not getting a green gumball) = 500/600 = 5/6 = .83333 = 83.3333%

Geometric Probability

Hint: Use Area! Then the rest is set up like our previous problems.

Area for Section A: ½ * 5*5 = 12.5

Area for Section B: 36 - 12.5 = 23.5

Area for Entire Section: 6*6 = 36

1) 12.5/36 = .3472 = 34.72%

2) 100 % - 34.72% = 65.28%

3) 100 %

4) 0 %

Event

Experimental Probabilities

Geometric Probability

Simple Event

Theoretical Probability

Fair Coin. Fair Die

Desired OutcomeOutcome

Closure

• What is the difference between geometric, experimental, and theoretical probabilities and give an example of each.

• Homework

Homework HelpGrade Eat Pizza Eat Burgers Eat Both Total9 5 10 15 3010 20 25 30 7511 35 40 45 12012 50 55 60 165Total 110 130 150 390

Factors of 4?Multiples of 3?Perfect square?Prime Number?Composite Number?

Probability ShowdownNumber your paper from

1-11You have 30 seconds to find the probability of the following situations. Answers must

be in simplest form.

1. A glass jar contains a total of 36 marbles. The jar has green and yellow marbles. There are 30 green marbles. What is probability of picking a green marble?

2. A glass jar contains a total of 31 marbles. The jar has purple and green marbles. There are 24 purple marbles. What is the probability of picking a green marble?

3. A bag contains 5 blue marbles, 5 green marbles, 3 purple marbles, 5 red marbles, and 5 yellow marbles. What is the probability of pulling out a blue or a green marble?

4. A number cube has 6 sides. The sides have the numbers 4, 8, 9, 9, 1, and 3. If the cube is thrown once, what is the probability of rolling a prime number?

5. If one letter is chosen at random from the word electorcute, what is the probability that the letter chosen is the letter "c"?

6. What is the probability of choosing a vowel out of the alphabet?

7. A pair of dice is rolled. What is the probability of getting a sum of 2?

8. In a class of 30 students, there are 17 girls and 13 boys. Five are A students, and three of these students are girls. If a student is chosen at random, what is the probability of choosing a girl or an A student?

9. A card is chosen from a standard deck of cards (no jokers). What is the probability that the card chosen is a red card?

10. A coin is flipped twice. What is the probability of flipping two heads?

11. Probability of landing in circle?

Answer #1

5/6

1. A glass jar contains a total of 36 marbles. The jar has green and yellow marbles. There are 30 green marbles. What is probability of picking a green marble?

Answer #2

7/31

2. A glass jar contains a total of 31 marbles. The jar has purple and green marbles. There are 24 purple marbles. What is the probability of picking a green marble?

Answer #3

10/23

3. A bag contains 5 blue marbles, 5 green marbles, 3 purple marbles, 5 red marbles, and 5 yellow marbles. What is the probability of pulling out a blue or a green marble?

Answer #4

1/6

4. A number cube has 6 sides. The sides have the numbers 4, 8, 9, 9, 1, and 3. If the cube is thrown once, what is the probability of rolling a prime number?

Answer #5

2/11

5. If one letter is chosen at random from the word electorcute, what is the probability that the letter chosen is the letter "c"?

Answer #6

5/26

6. What is the probability of choosing a vowel out of the alphabet?

Answer #7

1/36

7. A pair of dice is rolled. What is the probability of getting a sum of 2?

Answer #8

19/30

8. In a class of 30 students, there are 17 girls and 13 boys. Five are A students, and three of these students are girls. If a student is chosen at random, what is the probability of choosing a girl or an A student?

Answer #9

1/2

9. A card is chosen from a standard deck of cards (no jokers). What is the probability that the card chosen is a red card?

Answer #10

1/4

10. A coin is flipped twice. What is the probability of flipping two heads?

Answer #11

11. Landing in Circle

Area of Circle = pie*r2

Area of Square = L*W = 2r*2r =4r2

So, pie*r2

4r2

= Pie

4