Understanding the Multiplication Rule of Probability

Event $A$	The DNA from a crime scene is concluded to be the DNA of the defendant.
Event $B$	The defendant is not guilty.

Event

A

The DNA from a crime scene is concluded to be the DNA of the defendant.

Event

B

The defendant is not guilty.

$P (positive \| infected)$	$P (infected \| positive)$
$0.97$	$0.5$
$P (positive \| not infected)$	$P (not infected \| positive)$
$0.0001$	$0.5$

P (positive | infected)

P (infected | positive)

0.97

0.5

P (positive | not infected)

P (not infected | positive)

0.0001

0.5

Ali loves Russian authors. He has bought five books by Anton Chekhov and three books by Fyodor Dostoevsky. When going on vacation he randomly selects three books and throws them into his luggage. What is the probability that all three books are written by the same author? Round your answer to the nearest percent.

Let's represent books by Anton Chekhov with red covers, and represent books by Fyodor Dostoevsky with blue covers.

Ali has bought a total of 5+3=8 books. Since 5 of them are by Chekhov and 3 are by Dostoevsky, we get the following probabilities when Ali is choosing the first book. P(📕)=5/8 and P(📘)=3/8 When the first book is selected, there is one less book on the shelf. Depending on what book was selected, the probability of the second book Ali selects is affected. Let's mark the probabilities along the paths where three books by the same author, Chekhov or Dostoevsky, are selected.

To calculate the probability of either event occurring, we will use the Multiplication Rule of Probability. P(📕,📕,📕)=5/8* 4/7* 3/6=60/336 [1em] P(📘,📘,📘)=3/8* 2/7* 1/6=6/336 If we add these probabilities, we get the combined probability of selecting three books by the same author.

Ignacio was window shopping online when he found a shirt he really liked on sale. He hurried to order it and complete the purchase but forgot to specify the color. Because the color was not specified, the supplier will select the shirt color randomly. Here is a list of the two colors of shirts in stock for each supplier at the time of the purchase.

Supplier A	Supplier B
$30$ Blue	$40$ Blue
$10$ Red	$60$ Red

If Supplier A is currently taking $40 %$ of the orders, what is the probability that Ignacio gets a blue shirt? Make a tree diagram to find this probability and write the answer as a fraction in its simplest form.

To answer the question it will be helpful to make a tree diagram. The root node of the tree diagram represents the event of choosing a supplier. It can either be Supplier A or Supplier B. We also know that the probability that Supplier A has taken the orders is 40 %= 25. Let's start making the tree diagram using this information.

Recall that the sum of the probabilities of branches originating from one node is always 1. Using this fact, this part of tree diagram can be completed.

The next outcomes correspond to shirt colors.

Recall that Supplier A has 30 blue shirts and 10 red shirts. With this information, the probability that the shirt is of a given color can be calculated. P(Blue shirt|Supplier A)& = 30/30 + 10 = 30/40 = 3/4 [1em] P(Red shirt|Supplier A)& = 10/30 + 10 = 10/40 = 1/4 On the other hand, Supplier B has 40 blue shirts and 60 red shirts. With this information the probability that the shirt is of a given color can be calculated. P(Blue shirt|Supplier B)& = 40/40 + 60 = 40/100 = 2/5 [1em] P(Red shirt|Supplier B)& = 60/40 + 60 = 60/100 = 3/5 Let's add these probabilities to the diagram.

Finally, notice that we need to consider both the probability of getting a blue shirt from Supplier A P(Blue⋂A) or from Supplier B P(Blue⋂B).

Therefore, the probability that Ignacio receives a blue shirt is the sum of the probabilities that he receives the blue shirt from Supplier A or from Supplier B.

Therefore, the probability of receiving a blue shirt is 2750.

Jordan is getting ready for a party and wants to wear one of her dresses for the occasion. However, she does not want to worry about which one to pick, so she chooses one randomly. Consider the list of colors she has for the following articles of clothing.

Dresses	Pairs of Shoes
$2$ Blue	$1$ Black
$3$ Red	$1$ white

When she wears a blue dress,

4

out of

5

times she wears white shoes. On the other hand, when she wears a red dress she only wears white shoes

2

out of

5

times. What is the probability that she wears a blue dress and white shoes for the party? Use a tree diagram to find the probability and write the answer as a fraction in its simplest form.

To answer the question, a tree diagram will be made. The root node of the tree diagram represents the event of choosing a dress. It can either be blue or red.

The probability of each outcome should be written on the corresponding branch. Jordan has 2 blue dresses and 3 red dresses. With this information the probability that the dress is of a given color can be calculated. P(blue dress)&=2/2+ 3=2/5 [1em] P(red dress)&=3/2+ 3=3/5 Now, these probabilities will be added to the diagram.

The next outcomes correspond to the colors of shoes.

We know that the probability that Jordan wears white shoes with a blue dress is 45. The probability that Jordan wears white shoes with a red dress is 25. Let's add these probabilities to the diagram.

Recall that the sum of the probabilities of branches coming out of the same node is always 1. Using this fact, the tree diagram can be completed.

The probability that Jordan is wearing a blue dress and white shoes, can be calculated by multiplying the probabilities along the corresponding branches of the tree diagram.

Therefore, the probability that Jordan wears a blue dress and white shoes can be calculated as follows. 2/5*4/5 = 8/25

Jordan doesn't want to worry about what she will wear to go out. She picks a shirt, a pair of trousers, and a pair of shoes randomly. Consider the list of colors she has for each article of clothing.

Shirts	Trousers	Pair of shoes
$2$ Blue	$4$ Blue	$1$ Black
$3$ Red	$2$ Red	$2$ Violet
$2$ Pink	$1$ Black

What is the probability that she's wearing a blue shirt, black trousers, and violet shoes? Use a tree diagram to find the answer. Write your answer in exact form.

To answer the questions, a tree diagram will be made. The root node of the tree represents the event of choosing a shirt of a given color. It can either be blue, red, or pink.

The probability of each outcome should be written on the corresponding branch. Jordan has 2 blue shirts, 3 red ones, and 2 orange ones. With this information the probability that the shirt is of a given color can be calculated. P(Blue shirt)&=2/2+ 3+ 2=2/7 [1em] P(Red shirt)&=3/2+ 3+ 2=3/7 [1em] P(Pink shirt)&=2/2+ 3+ 2=2/7 Now, the probabilities will be added to the diagram.

The next outcomes is the color choice for the trousers.

Jordan has 3 blue trousers, 2 red trousers, and 1 black trouser ones. With this information the probability that the trouser is of a given color can be calculated. P(Blue trousers)&=3/3+ 2+1=3/6=1/2 [1em] P(Red trousers)&=2/3+ 2+1=2/6=1/3 [1em] P(Black trousers)&=1/3+ 2+1=1/6 Now, the probabilities will be added to the diagram.

Finally, it is left to consider the possible outcomes for the color choice of the pair of shoes.

Jordan has 1 pair of black shoes and 2 pairs of violet shoes. With this information the probability that the pair of shoes chosen is of a given color can be calculated. P(Black shoes)&=1/1+ 2=1/3 [1em] P(Violet shoes)&=2/1+ 2=2/3 Now, the probabilities will be added to the diagram.

The probability that Jordan is wearing a blue shirt, black trousers, and violet shoes can be calculated by multiplying the probabilities along the corresponding branch of the tree diagram.

Therefore, the probability that Jordan wears a blue shirt, black trousers, and violet shoes is P = 2/7*1/3*2/3 = 4/63

How a conditional probability can be reversed was taught during the lesson, arriving at the formula shown below.

P (A ∣ B) = \frac{P ( B ∣ A ) \cdot P ( A )}{P ( B )}

This result is known as Bayes' Theorem, and it is very useful when working with conditional probabilities problems. Consider the following situation. Kriz wants to travel to visit their family. They find two options from two different airlines that offer affordable prices. However, according to customers reviews, they both have high chances of being delayed.

Airline A	Airline B
On time $30 %$	On time $20 %$
Delayed $70 %$	Delayed $80 %$

Since Kriz can only afford a ticket from one of these airlines, they decide to give it a try. They put

4

pieces of paper reading Airline A and

6

reading Airline B to make the decision by selecting one piece of paper randomly. On the flight day, Kriz arrived on time. What is the probability that Kriz traveled with Airline B? Write the answer in exact form.

Since we know that Kriz used 4 pieces of paper reading Airline A and 6 reading Airline B to make their decision, we can calculate the probability that they have flown with a given airline. P(airline A) &= 4/4+ 6 = 4/10 = 2/5 [1em] P(airline B) &= 6/4+ 6 = 6/10 = 3/5 Let's summarize all the known probabilities so far.

Flying with Airline A	Arriving late given that they traveled with Airline A	Arriving on time given that they traveled with Airline A
P(airline A) = 2/5	P(late \|airline A) = 70 % = 7/10	P(on time \|airline A) = 30 % = 3/10
Flying with Airline B	Arriving late given that they traveled with Airline B	Arriving on time given that they traveled with Airline B
P(airline B) = 2/5	P(late \|airline B) = 80 % = 4/5	P(on time \|airline B) = 20 % = 1/5

We are interested in finding the probability that Kriz traveled with Airline B, given that they arrived on time. Notice that the probability that Kriz arrived on time provided that they flew with Airline B is known. cc Desired & Given P(airline B|on time) & P(on time|airline B) To reverse the given conditional probability we can use Bayes' Theorem, as mentioned at the beginning of the exercise. P(airline B|& on time) & = P(on time | airline B)* P(airline B)/P(on time) All the probabilities on the right-hand side of the equation are known, except for P(on time). To calculate it, let's represent the situation using a tree diagram.

We need to consider both the probability of arriving on time given that the flight was with Airline A and the probability of arriving on time given that the flight was with Airline B. These can be obtained by multiplying the probabilities along the corresponding branches.

Therefore, the probability that the flight arrives on time P(on time) is the sum of the probability that it arrives on time given that the flight was with Airline A and the probability that it arrives on time given that the flight was with Airline B.

Now, we can go back to the expression for P(airline B | on time) and find the required probability.

Therefore, the probability that Kriz had used Airline B given that they arrived on time is 13.

Multiplication Rule in the Uniform Probability Model

Catch-Up and Review

Prosecutor's Fallacy

Investigating Conditional Probability

Multiplication Rule of Probability

Proof

Using a Tree Diagram to Determine Probability

Hint

Solution

Finding Probabilities Along Branches of a Tree Diagram

Using a Tree Diagram to Determine Probability

Hint

Solution

Reversing Conditional Probability

Hint

Solution

Practice Reversing Conditional Probability

Investigating Conditional Probability

Solution

Probability That a Person Is Infected if the Test Is Negative

Probability That a Person Is Infected if the Test Is Positive

Probability That a Person Is Not Infected if the Test Is Positive

Extra

Multiplication Rule in the Uniform Probability Model

Recommended exercises

	10 Theory slides
	8 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions