Basics of Probability: Outcomes, Events, and Uniform Probability Model

	Statement (Probability of)
A	Drawing a card with the number $14$ from a standard deck
B	Independence Day will be on July $4^{th}$
C	A coin toss resulting in heads
D	Being born on leap day
E	Getting a number greater than $1$ on the roll of a standard die

Statement (Probability of)

Drawing a card with the number

14

from a standard deck

Independence Day will be on July

4^{th}

A coin toss resulting in heads

Being born on leap day

Getting a number greater than

1

on the roll of a standard die

Drawing a card with the number $14$ from a standard deck

Independence Day will be on July $4^{th}$

A coin toss resulting in heads

Being born on leap day

Getting a number greater than $1$ on the roll of a standard die

	Statement (Probability of)	Likelihood
A	Drawing a card with the number $14$ from a standard deck	Will not happen
B	Independence Day will be on July $4^{th}$	Will happen
C	A coin toss resulting in heads	Equally likely to happen or not happen
D	Being born on leap day	Unlikely to happen
E	Getting a number greater than $1$ on the roll of a standard die	Likely to happen

Statement (Probability of)

Likelihood

Drawing a card with the number

14

from a standard deck

Will not happen

Independence Day will be on July

4^{th}

Will happen

A coin toss resulting in heads

Equally likely to happen or not happen

Being born on leap day

Unlikely to happen

Getting a number greater than

1

on the roll of a standard die

Likely to happen

Game (Experiment)	How to Win (Event)	Probability of Winning
Randomly picking a number from the set formed by the numbers from $1$ to $20$	Picking a multiple of $4$	$\frac{1}{4}$
Randomly drawing a card from a standard deck of cards	Drawing a red card	$\frac{1}{2}$
Spinning a $3 -$ colored roulette wheel evenly divided	Landing on blue	$\frac{1}{3}$
Randomly selecting a letter from the English Alphabet	Picking the letter M	$\frac{1}{2 6}$

Game (Experiment)

How to Win (Event)

Probability of Winning

Randomly picking a number from the set formed by the numbers from

1

20

Picking a multiple of

4

\frac{1}{4}

Randomly drawing a card from a standard deck of cards

Drawing a red card

\frac{1}{2}

Spinning a

3 -

colored roulette wheel evenly divided

Landing on blue

\frac{1}{3}

Randomly selecting a letter from the English Alphabet

Picking the letter M

\frac{1}{2 6}

$P (Theme Park Passes)$
Experimental Probability	Theoretical Probability
$\frac{3}{2 0} = 0.15$	$\frac{1}{6} = 0.1 \overline{6}$

P (Theme Park Passes)

Experimental Probability

Theoretical Probability

\frac{3}{2 0} = 0.15

\frac{1}{6} = 0.1 \overline{6}

Flipping a Coin and Rolling a Die
Possible Outcomes
$(H, 1)$	$(H, 2)$	$(H, 3)$	$(H, 4)$	$(H, 5)$	$(H, 6)$
$(T, 1)$	$(T, 2)$	$(T, 3)$	$(T, 4)$	$(T, 5)$	$(T, 6)$

Flipping a Coin and Rolling a Die

Possible Outcomes

(H, 1)

(H, 2)

(H, 3)

(H, 4)

(H, 5)

(H, 6)

(T, 1)

(T, 2)

(T, 3)

(T, 4)

(T, 5)

(T, 6)

Choose the appropriate answer that describes all outcomes of the following experiments.

The result of a soccer game between Arsenal and Real Madrid.

Selecting a fruit from a basket containing pears, mangoes, strawberries, and oranges.

When conducting a probability experiment, an outcome refers to a possible result. In this experiment, we want to determine the potential outcomes of a soccer game between Arsenal and Real Madrid.

In this case, there are three possibilities — Arsenal wins, Real Madrid wins, or they draw. Outcomes: {Arsenal wins, Real Madrid wins, draw} It is worth mentioning that the other options do not describe all outcomes. For instance, Arsenal winning and Real Madrid losing represent the same outcome. Also, Real Madrid winning and Arsenal losing represents only one outcome.

Our goal is to identify all the possible outcomes of selecting a fruit from a basket that only consists of pears, mangoes, strawberries, and oranges.

It is worth noting that the quantity of each fruit in the basket is irrelevant. The only available options for selection are the fruits that are present in the basket, namely a pear, a mango, a strawberry, or an orange. Outcomes: {pear, mango, strawberry, orange}

Classify each event as one of the following: likely to happen, unlikely to happen, will happen, will not happen, or equally likely to happen or not happen.

Rolling a fair six-side die and getting a number less than

7 .

Drawing a red marble from a bag containing

5

red marbles and

40

marbles in total.

Picking a random number between

1

and

10

and having it be both odd and even.

Consider the given event.

Rolling a fair six-side die and getting a number less than 7.

A standard die has six sides with numbers ranging from 1 to 6.

We can guarantee that the result will be a number less than 7 as all possible outcomes are below 7. Therefore, this event will happen.

Let's begin by looking at the given event.

Drawing a red marble from a bag containing 5 red marbles and 40 marbles in total.

We have a bag containing 40 marbles, of which 5 are red. This implies that 35 marbles are of some other color. The probability of drawing a red marble is low because there are more non-red marbles in the bag. Therefore, this event is unlikely to happen.

Consider the given event.

Picking a random number between 1 and 10 and having it be both odd and even.

A number can only be either odd or even, but not both at the same time. This means it is impossible to find a number in the range 1--10 that is both even and odd. As a result, this event will not happen.

Determine the likelihood of the following scenarios. Write the answer in the form of a simplified fraction.

Selecting a month at random from the year and it turning out to be July.

Drawing a card randomly from a standard deck and it being a face card (either a Jack, Queen, or King).

Picking a random number between $1$ and $30$ and it being a multiple of $3 .$

Look at the given event.

Selecting a month at random from the year and it turning out to be July.

We want to determine the probability of randomly choosing July from the twelve months in a year. January, February, March, April May, June, July, August September, October, November, December Since July is only one of the twelve potential outcomes, the probability of selecting it is represented by the ratio 1: 12. Let's write this ratio as a fraction. P(July)=1/12

Consider the given event.

Drawing a card randomly from a standard deck and it being a face card (either a Jack, Queen, or King).

A standard deck has four suits. Each suit has three face cards — a jack, a queen, and a king. The product of the face cards per suit and the number of suits gives the number of favorable outcomes. Number of favorable outcomes= 3* 4 ⇕ Number of favorable outcomes= 12 A standard deck has a total of 52 cars. Therefore, the ratio of 12 to 52 gives the probability of drawing a face card from a deck of cards. P(Face card)=12/52 ⇕ P(Face card)=3/13

Begin by looking at the given event.

Picking a random number between 1 and 30 and it being a multiple of 3.

We can create a list of all the multiples of 3 that are less than or equal to 30. Afterwards, we can count the number of favorable outcomes for this event. Multiples of3:& 3,6,9,12,15,18, & 21,24,27,30 Out of the total of 30 possible outcomes, there are 10 multiples of 3, which is the number of favorable outcomes. We can express the probability of obtaining a multiple of 3 as a ratio of 10 to 30. P(Multiple of3)=10/30 ⇕ P(Multiple of3)=1/3

Look at the following events.

A . B . C . A friend selects a random month and it is a month with exactly 30 days . A friend picks a random number from 1 to 13 that is a prime number A friend draws a card from a standard deck and it is a red queen .

Match each event with its corresponding probability.

We will examine each event individually to determine its probability. To do so, we will identify all possible outcomes and the favorable outcomes for each event.

Event A

Let's look at event A.

A friend selects a random month and it is a month with exactly 30 days.

We are interested in determining the probability of selecting a month with precisely 30 days. We know that there are 12 months, which is the total number of possible outcomes. Now, let's write a list of every month and its corresponding number of days.

Month	Number of Days
January	31
February	28 in a common year and 29 in a leap year
March	31
April	30
May	31
June	30
July	31
August	31
September	30
October	31
November	30
December	31

We can see that only 4 months have exactly 30 days. This number represents the total number of favorable outcomes. Let's write the ratio of the number of favorable outcomes to the number of possible outcomes to determine the probability of picking a month with precisely 30 days. P(A)=4/12 ⇔ P(A)=1/3

Event B

Let's focus on event B.

A friend picks a random number from 1 to 13 that is a prime number.

We want to calculate the probability of randomly choosing a prime number from 1 to 13. A prime number is a number that only has 1 and itself as factors. Let's make a list of all the numbers from 1 to 13 and highlight the prime numbers. 1, 2, 3,4, 5,6, 7, 8,9,10, 11,12, 13 Out of the 13 possible outcomes, the 6 highlighted numbers are the favorable ones. Let's write the ratio of the favorable outcomes to the possible outcomes to determine the probability of event B. P(B)=6/13

Event C

Consider event C.

A friend draws a card from a standard deck that is a red queen.

There are 52 cards in a standard deck, and each suit has one queen. Out of the four queens, 2 are red — hearts and diamonds. Number of favorable outcomes= 2 Number of possible outcomes= 52 We can write the ratio of the number of favorable outcomes to the number of possible outcomes to find the probability of event C. P(C)=2/52 ⇔ P(C)=1/26

Conclusion

We have determined the probability of each of the given events. Let's summarize our results in a table.

Event	Description	Probability
A	A friend selects a random month and it is a month with exactly 30 days.	1/3
B	A friend picks a random number from 1 to 13 that is a prime number.	6/13
C	A friend draws a card from a standard deck that is a red queen.	1/26

Heichi has $58$ books in his collection, with $6$ being fantasy and $8$ being science.

If he chooses a book at random, what is the probability that he will select a book that is neither a fantasy nor a science? Write the answer in a simplified fraction.

We need to determine the probability of Heichi selecting a book that is not a fantasy or science. The probability of not choosing a fantasy or science book is the opposite of choosing one. This means that the sum of these complementary events adds up to 1. P(Not choosing a fantasy or science book) + P(Choosing a fantasy or a science book) =1 We know that out of the 58 books, 6 are fantasy and 8 are science. That means we have a total of 6+8= 14 books that fall into either of these categories. We can use the ratio of 14 to 58 to find the probability of choosing a fantasy or science book. P(Choosing a fantasy or science book)=14/58 ⇕ P(Choosing a fantasy or science book)=7/29 We can now substitute this ratio into the equation written previously for the sum of the complementary events. P(Not choosing a fantasy or science)+ 7/29=1 We now need to find the number that added to this ratio equals 1. This number is 2229. P(Not choosing a fantasy or science)+ 7/29=1 ⇓ 22/29+ 7/29=1 We can then infer that the probability of Heichi not choosing a fantasy or science book is 2229.

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

The Basics of Probability

Catch-Up and Review

Hint

Solution

Option A

Option B

Option C

Option D

Option E

Conclusion

Hint

Solution

Hint

Solution

Hint

Solution

Picking a Multiple of 4

Drawing a Red Card

Landing on Blue

Picking the Letter M

Conclusions

Hint

Solution

Event A

Event B

Event C

Conclusion

The Basics of Probability

Recommended exercises

Picking a Multiple of $4$