Two blue marbles ...or dependent and independent probability

found my marbles.... by mell242

Design an experiment using coins to simulate a 10 question true/false test. What is the experimental probability of scoring at least 70% on the test if you guess each answer?

Let's think about this again using what we've just learned ...

Solve for the exact theoretical probability of getting "at least 7" out of ten on this test.

HOMEWORK

(a) How many numbers of 5 different digits each can be formed from the digits 0, 1, 2, 3, 4, 5, 6?

(b) If one of these numbers is randomly selected, what is the probability it is even?

(c) What is the probability it is divisible by 5?

HOMEWORK

Seven people reach a fork in a road. In how many ways can they continue their walk so that 4 go one way and 3 the other?

HOMEWORK

You write a 10 question multiple choice test with 4 choices possible for each question. What is the probability of scoring at least 10% on the test if you guess each answer?

Dependent and independent probabilities ...

A bag contains 6 marbles, 3 red and 3 blue. A marble is chosen at random and then replaced back in the bag. A second marble is selected, what is the probability that it is blue?

Independent EventsEvents in which the outcome of one event does not affect the outcome of the other event.

Example

Dependent and independent probabilities ...

A bag contains 6 marbles, 3 red and 3 blue. A marble is chosen at random and NOT replaced back in the bag. A second marble is selected, what is the probability that it is blue?

Dependent EventsIf the outcome of one event affects the outcome of another event, then the events are said to be dependent events.Example

Which of the following are dependent events? Which are independent?

Examples:

1. Removing (selecting randomly) three red marbles without replacement from a bag that contains six red and nine blue marbles.

2. Selecting a red card from a deck of cards, returning the card to the deck, shuffling the cards, and selecting a second red card.

3. Rolling two dice.

4. The weather and how likely you are to go visiting. You have decided that there is a 50% chance that you will visit your friend if it does not snow, and a 10% chance if it does snow.

MarblesA jar contains 5 red and 7 blue marbles. What is the probability of pulling out 2 blue marbles in a row, with replacement?

Marbles (again)A jar contains 5 red and 7 blue marbles. What is the probability of pulling out 2 blue marbles in a row, without replacement?

Breakfast for RupertRupert has either milk or cocoa to drink for breakfast with either oatmeal or pancakes. If he drinks milk, then the probability that he is having pancakes with the milk is 2/3. The probability that he drinks cocoa is 1/5. If he drinks cocoa, the probability of him having pancakes is 6/7.

a) Show the sample space of probabilities using a tree diagram or any other method of your choice.

b) Find the probability that Rupert will have oatmeal with cocoa tomorrow morning.

pl

Testing for independence ...

30% of seniors get the flu every year. 50% of seniors get a flu shot annually. 10% of seniors who get the flu shot also get the flu. Are getting a flu shot and getting the flu independent events?

P(shot) = 0.50P(flu) = 0.30P(shot & flu) = (0.50)(0.30) = 0.15

HoweverP(shot & flu) = 0.10

The probability that Gallant Fox will win the first race is 2/5 and that Nashau will win the second race is 1/3.

3. What is the probability that at least one horse will win a race?

2. What is the probability that both horses will lose their respective races?

1. What is the probability that both horses will win their respective races?

Mutually Exclusive Events ...

Two events are mutually exclusive (or disjoint) if it is impossible for them to occur together.

Formally, two events A and B are mutually exclusive if and only if

Examples:1. Experiment: Rolling a die once Sample space S = {1,2,3,4,5,6} Events A = 'observe an odd number' = {1,3,5} B = 'observe an even number' = {2,4,6} A ∩ B = ∅ (the empty set), so A and B are mutually exclusive.

2. A subject in a study cannot be both male and female, nor can they be aged 20 and 30. A subject could however be both male and 20, or both female and 30.

Mutually ExclusiveNot Mutually Exclusive

A B1 2

3 45 6

A B

ExampleSuppose we wish to find the probability of drawing either a king or a spade in a single draw from a pack of 52 playing cards.

We define the events A = 'draw a king' and B = 'draw a spade'Since there are 4 kings in the pack and 13 spades, but 1 card is both a king and a spade, we have:

P(A U B) = P(A) + P(B) - P(A ∩ B) = 4/52 + 13/52 - 1/52 = 16/52

P(A and B) = P(A ∩ B) = P(A) * P(B) = (4/52) * (13/52) = 1/52

So, the probability of drawing either a king or a spade is 16/52 = 4/13.

Probabilities involving "and" and "or" A.K.A "The Addition Rule"...

The addition rule is a result used to determine the probability that event A or event B occurs or both occur.

The result is often written as follows, using set notation:

P(A or B) = P(A∪B) = P(A)+P(B) - P(A∩B)

where: P(A) = probability that event A occurs P(B) = probability that event B occurs P(A U B) = probability that event A or event B occurs P(A ∩ B) = probability that event A and event B both occur

P(A and B) = P(A∩B) = P(A)*P(B)

A BA B

A∪B A∩Bor and

Identify the events as mutually exclusive, or not mutually exclusive.

a. A bag contains four red and seven black marbles. The event is randomly selecting a red marble from the bag, returning it to the bag, and then randomly selecting another red marble from the bag.

b. One card - a red card or a king - is randomly drawn from a deck of cards.

c. A class president and a class treasurer are randomly selected from a group of 16 students.

d. One card - a red king or a black queen - is randomly drawn from a deck of cards.

e. Rolling two dice and getting an even sum or a double.

Chad has arranged to meet his girlfriend, Stephanie, either in the library or in the student lounge. The probability that he meets her in the lounge is 1/3, and the probability that he meets her in the library is 2/9.

a. What is the probability that he meets her in the library or lounge?

b. What is the probability that he does not meet her at all?

The probability that Tony will move to Winnipeg is 2/9, and the probability that he will marry Angelina if he moves to Winnipeg is 9/20. The probability that he will marry Angelina if he does not move to Winnipeg is 1/20. Draw a tree diagram to show all outcomes.

1. What is the probability that Tony will move to Winnipeg and marry Angelina?

3. What is the probability that Tony does not move to Winnipeg and does not marry Angelina?

2. What is the probability that Tony does not move to Winnipeg but does marry Angelina?

HOMEWORK

(a) How many different 4 digit numbers are there in which all the digits are different?

(b) If one of these numbers is randomly selected, what is the probability it is odd?

(c) What is the probability it is divisable by 5?

HOMEWORK

An examination consists of thirteen questions. A student must answer only one of the first two questions and only nine of the remaining ones. How many choices of questions does the student have?

HOMEWORK

Randomly arranged on a bookshelf are 5 thick books, 4 medium-sized books, and 3 thin books. What is the probability that the books of the same size stay together?

HOMEWORK