4.2 to 4.5 Probabilities Intro, Conditional, Mutually Exclusive,

​4.2 to 4.5: Probability, Random Variables, and Probability Distributions

By Jeffrey Reed

• ​4.2 Estimating Probabilities

• 4.3 Introduction to Probability

• 4.4 Mutually Exclusive Events

• 4.5 Conditional Probability

Draw

Shade the appropriate area.

Draw

Shade the appropriate area.

Multiple Choice

The diagram shows the sports played by 80 students.

If a student is picked at random, what is the probability that they play all three sports?

• ³¹/₈₀

• ⁴⁵/₈₀

• ⁶²/₈₀

• ⁶⁷/₈₀

Multiple Choice

The diagram shows the sports played by 80 students.

If a student is picked at random, what is the probability that they play football only?

• ⁶⁶/₈₀

• ⁴/₈₀

• ¹⁸/₈₀

• ⁴⁹/₈₀

Multiple Choice

Identify the shaded area...

• A∩B

• A∪B

• A'∩B'

• A'∪B'

Mutually exclusive events are events that cannot happen at the same time. The probability of two mutually exclusive events, A and B, occurring simultaneously is zero: P(A and B) = 0.​


Not mutually exclusive events, also called inclusive events, are events that can occur at the same time because they have overlapping outcomes.

Conditional probability is the likelihood of an event occurring, given that another event has already occurred. It is written as P(A|B), meaning the probability of event A happening given that event B has already happened.

Multiple Choice

You are playing a game using this spinner. You get one spin on each turn. Find a complete probability model for one spin of the spinner.

• P(number ​3)=1/4, ​P(letter ​A)= 1/4

• P(number)=1/2​, ​P(letter ​Y)= 1/4

• P(letter)=1/2​, P(number)=1/2

• P(letter)=1/2​, ​P(number ​5)= 1/4

Multiple Choice

What are the two parts that describe a probability model?

• Sample space and outcomes

• Sample space and probability

• Probability and chance behavior

• Outcomes and simulations

Multiple Choice

What is the addition rule for mutually exclusive events?

• P(A) + P(B) = P(A or B)

• P(A) + P(B) = P(A and B)

• P(A) + P(B) = 1

• P(A) + P(B) = P(A) - P(B)

Multiple Choice

While preparing lessons for Distance Learning, your teachers are eating their Easter candy. Suppose a piece of candy is selected at random from those represented in the table. Find P( $$Mrs\ T^c$$ ). 

• $$\frac{50}{149}$$  

• $$\frac{24}{50}$$  

• $$\frac{99}{149}$$  

• $$\frac{6}{50}$$  

Multiple Choice

The question “Do you play soccer?” was asked of 110 students. Results are shown in the table.

19) Given a student is female, what is the probability the student plays soccer?

• 12/54

• 42/54

• 12/35

• 42/75

Multiple Choice

Which of the following shows how to determine when a die is rolled P(multiple of 2 or a multiple of 3)?

• 2/6 + 1/6 - 3/6
• 4/6 + 2/6 - 1/6
• 3/6 + 2/6 - 1/6
• 5/6 + 1/6 - 3/6

Multiple Choice

What is the intersection of events A and B represented as in a Venn diagram?

• A ∩ B

• A U B

• A

• B

Open Ended

In a bag of M&Ms, the probability of randomly picking a blue M&M, written P(blue) is 0.207. Using the complement rule, what is P(blue')?

Multiple Choice

In a landfill, there is a 27% chance that a piece of trash is recyclable plastic, a 54% chance that the item is biodegradable, and a 9% chance that the item is neither of these things. Pick of piece of trash at random. What is the probability that it is recyclable plastic or biodegradable?

• 81%

• 21%

• 54%

• 63%

Poll

How confident do you feel about this topic now?

• Very confident
• Somewhat confident
• Not confident

Math Response

Answer using a non-simplified fraction.

P(bass | 7th grade)

Type answer here

Correct Answer: \frac{16}{60}

Deg^°

Rad

​Conditional probability is the likelihood of an event occurring, given that another event has already occurred. It is written as P(A|B), meaning the probability of event A happening given that event B has already happened.
Mutually exclusive events are events that cannot happen at the same time. The probability of two mutually exclusive events, A and B, occurring simultaneously is zero: P(A and B) = 0.

Not mutually exclusive events, also called inclusive events, are events that can occur at the same time because they have overlapping outcomes.

Multiple Choice

Marti rolls two dice. What is the probability that the sum of the dice is equal to 7, given that the first die is showing a 2? Write as a simplified fraction using "/".

• 1/6
• 1/12

• 2/3

• 1/2

Multiple Choice

What is the probability a passenger on the Titanic surviving given they are in first class?

• .2855

• .6246

• .0922

• .3230

Open Ended

Are the events "male" and "left-handed" independent? Justify your answer.

Multiple Choice

A bottle contains 1 green and 1 red ball. A ball is drawn at random, replaced in the bottle and another ball is drawn. Find the sample space from the tree diagram shown.

• GR, RG, RR

• GG, RR

• GG, GR, RR

• GG, GR, RG, RR

Multiple Choice

What is the formula for finding the probability that events A and B both occur?

• P(A ∩ B) = P(A) + P(B | A)

• P(A ∩ B) = P(A) • P(B | A)

• P(A ∩ B) = P(A) - P(B | A)

• P(A ∩ B) = P(A) / P(B | A)

Multiple Choice

Use the tree diagram to find the probability of tossing a head first and then a tail when a coin is tossed twice.

• 1

• 1_/2

• 1/₄

• 1/8

Multiple Choice

What is the multiplication rule for independent events?

• P(A ∩ B) = P(A) + P(B)

• P(A ∩ B) = P(A) • P(B)

• P(A ∩ B) = P(A) - P(B)

• P(A ∩ B) = P(A) / P(B)

Multiple Choice

A game is played where a fair coin is flipped.

Heads you lose, tails you get to flip again.

On the second go heads you lose and tails you win.

What is the probability of winning?

• ¹/₂

• ¹/₄

• ³/₄

• 1

Multiple Choice

What will we learn about in the next chapter regarding random variables?

• Discrete and Continuous Random Variables

• Transforming and Combining Random Variables

• Binomial and Geometric Random Variables

• All of the above

Dropdown

FIRST BLANK

Select answer

SECOND BLANK ​ ​ ​

Select answer

Match

A school requires students to wear uniforms. They can choose from the following options.

• How many total choices does a student have?

• What is the probability of choosing a gray hoodie, navy polo and khaki pants?

• What is the probability of choosing a black polo?

16

1/16

1/4

• 16

• 1/16

• 1/4

Multiple Choice

If you are picking a card randomly from a deck of cards, the events of picking a jack and picking a heart are mutually exclusive.

• True

• False

Explanation Slide...

The statement is false because the Jack of Hearts is a card that is both a Jack and a Heart, meaning these two events are not mutually exclusive as they can occur at the same time. Two events are mutually exclusive if they cannot both happen at once, but since you can draw the Jack of Hearts, there is an overlap, and thus no mutual exclusivity. Why they are not mutually exclusiveDefinition of mutually exclusive events: Events are mutually exclusive if they cannot occur simultaneously. The overlap: A standard deck of cards contains a Jack of Hearts. Demonstration: This specific card fulfills both conditions: it is a Jack, and it is a Heart. Therefore, drawing a Jack and drawing a Heart can both happen if the card drawn is the Jack of Hearts. 

Multiple Choice

If you draw one card from a standard deck, what is the probability of drawing a 5 or a diamond?

• 2/52
• 4/52
• 16/52
• 26/52

Explanation Slide...

The probability of drawing a 5 or a diamond is4134 over 13 end-fraction413. Explanation: To calculate the probability of drawing a 5 or a diamond, you can use the formula for the probability of the union of two events:P(A∪B)=P(A)+P(B)−P(A∩B)cap P open paren cap A union cap B close paren equals cap P open paren cap A close paren plus cap P open paren cap B close paren minus cap P open paren cap A intersection cap B close paren𝑃(𝐴∪𝐵)=𝑃(𝐴)+𝑃(𝐵)−𝑃(𝐴∩𝐵). Step 1: Determine the total number of cards. A standard deck has 52 cards.Step 2: Find the number of cards that are a 5. There are four 5s (one for each suit).Step 3: Find the number of cards that are diamonds. There are 13 diamonds in a deck.Step 4: Identify the overlap. There is one card that is both a 5 and a diamond (the 5 of diamonds).Step 5: Calculate the total number of favorable outcomes by adding the number of 5s and the number of diamonds, and then subtracting the overlapping card to avoid double-counting:4+13−1=164 plus 13 minus 1 equals 164+13−1=16.Step 6: Divide the number of favorable outcomes by the total number of cards to get the probability:165216 over 52 end-fraction1652.Step 7: Simplify the fraction by dividing the numerator and denominator by 4:16÷452÷4=413the fraction with numerator 16 divided by 4 and denominator 52 divided by 4 end-fraction equals 4 over 13 end-fraction16÷452÷4=413. The probability is4134 over 13 end-fraction𝟒𝟏𝟑.

Multiple Choice

If you draw one card from a standard deck, what is the probability of drawing a spade or a red card?

• 13/52
• 26/52
• 39/52
• Not possible

Explanation Slide...

The probability of drawing a spade or a red card from a standard 52-card deck is 3/4 or 39/52. This is because there are 13 spades and 26 red cards (hearts and diamonds), and since spades are black, there is no overlap between the "spade" and "red" categories. Adding the number of favorable cards (13 spades + 26 red cards = 39) and dividing by the total number of cards (52) gives the probability. Step-by-step calculation:Total Cards: A standard deck has 52 cards. Number of Spades: There are 13 spades in the deck. Number of Red Cards: There are 26 red cards (13 hearts and 13 diamonds). Favorable Outcomes: The favorable outcomes are drawing a spade or a red card. Since spades are black, there's no overlap with the red cards. Total favorable cards = Number of spades + Number of red cards = 13 + 26 = 39. Calculate Probability: Probability = (Favorable Outcomes) / (Total Outcomes). Probability = 39 / 52. Simplify: The fraction 39/52 can be simplified to 3/4. Therefore, the probability of drawing a spade or a red card is 3/4. 

Multiple Choice

If events A and B are mutually exclusive, what is P(A and B)?

• 0.50

• 0

• 1

• 0.75

Explanation Slide...

If events are mutually exclusive, they cannot happen at the same time. The probability would be 0

Multiple Choice

• No.

• Yes.

• Maybe.

• I dont know.

Multiple Choice

If a student is chosen at random, find the probability of getting someone who is a regular or heavy smoker. Round your answer to three decimal places.

• 0.221

• 0.662

• 0.193

• 0.128

Poll

How confident do you feel about this topic now?

• Very confident
• Somewhat confident
• Not confident