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Consider the game darts. It is more likely — probable — to hit the larger areas than hitting the inner circles in the very middle. This is because the game is designed that way. A new player can then predict they will hit a larger area.
### Catch-Up and Review

However, there are situations where making predictions is more challenging. In such cases, *probability* can be used to arrive at informed decisions and determine the likelihood of events happening. These concepts will be explored in this lesson.

**Here are a few recommended readings before getting started with this lesson.**

Explore

Roll the die and record the results in the table.

Consider the following scenario. If the die is rolled $100$ times, what is the expected number of times each number will be rolled?

Discussion

Probability measures the likelihood that something will occur. It can be any value from $0$ to $1$ or from $0%$ to $100%,$ inclusive. When it is certain that the situation *will not* occur, the probability is $0.$ Likewise, when it is certain that the situation *will* occur, the probability is $1.$

Example

Kevin and Davontay want to finish their homework quickly so they can convince their parents to take them out for burgers. Their assignment requires them to categorize a set of statements based on their probability.

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 |

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Analyze each statement one at a time. Weigh the chances of the outcome happening against the chances of it not happening.

Each statement will be analyzed one at a time to determine its likelihood.

Look at the given statement.

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

A standard deck has $52$ cards divided into four suits — spades, clubs, hearts, and diamonds. Each suit has $14$ cards which include an ace, a $2,$ a $3,$ a $4,$ a $5,$ a $6,$ a $7,$ a $8,$ a $9,$ a $10$, a jack, a queen, and a king. Drawing a card with the number $14$ is impossible since no such card exists in the deck.

Now, consider the second statement.

Independence Day will be on July $4_{th}$ |

Independence Day is always celebrated on July $4_{th},$ so it is a guaranteed occurrence. It will happen.

Consider the third statement.

A coin toss resulting in heads |

When flipping a coin, there are only two possible outcomes — heads or tails. Because both outcomes have an equal chance of occurring, the likelihood of getting heads is the same as the likelihood of not getting heads.

Let's breakdown the given statement.

Being born on leap day |

Leap Day, February $29,$ is added to the calendar every four years, so it takes three years before it happens again. As a result, being born on Leap Day is unlikely to happen.

Consider the final statement.

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

A standard die has six sides with numbers from $1$ to $6.$ Getting a number greater than $1$ is likely to happen because most outcomes are greater than $1.$

All statements have been analyzed. The results are summarized in the table below.

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 |

Discussion

An experiment is a process used to determine the probability of an event occurring in the future. In finding probability, an experiment is an action that can be repeated infinitely many times and has a variety of results called outcomes. Even rolling a die can be considered an experiment. *trial*. For example, if a die is thrown $100$ times, each throw is considered a trial.

Experiments are repeated several times to collect data, and each repetition is called a

Discussion

An outcome is a possible result of a probability experiment. For example, when rolling a six-sided die, getting a $3$ is one possible outcome.

Note that when performing an experiment, each possible outcome is unique. That means only one outcome will occur for each trial.Discussion

An event is a combination of one or more specific outcomes. For example, when playing cards, an event might be drawing a spade or a heart. For this event, one possible outcome is drawing the $A♠$ or drawing the $7♡.$

However, these are not the only outcomes of this event. All the possible outcomes that satisfy the event are listed below.

$Outcomes:A♠,2♠,3♠,4♠,5♠,6♠,7♠8♠,9♠,10♠,J♠,Q♠,K♠A♡,2♡,3♡,4♡,5♡,6♡,7♡8♡,9♡,10♡,J♡,Q♡,K♡ $

The probability of an event is a ratio that compares the number of favorable outcomes to the number of possible outcomes.

$P(event)=Number of possible outcomesNumber of favorable outcomes $

Example

Davontay and Kevin finish their homework, and it is now time to order burgers! They each ordered a happy meal. Davontay choose the meal set that gives away a toy car. He hopes to get a Twin Mill Hot Wheel. Kevin, on the other hand, ordered a meal that offers a book. He hopes to get the book The Sparkle Tales

.

a If there are eight different Hot Wheels with an equal chance of getting each one, what is the probability of Davontay getting the Twin Mill Hot Wheel?

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b If there are five books with an equal chance of getting either one, what is the probability of Kevin getting the book he wants?

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a What is the number of different Hot Wheels that Davontay can get? And out of those, how many are Twin Mill? Write the ratio of the number of favorable outcomes to the number of possible outcomes.

b Identify the number of different books that Kevin can get. How many of these are

The Sparkle Tales? Calculate the ratio of the number of favorable outcomes to the number of possible outcomes.

a Davontay can get one of eight possible Hot Wheels in his happy meal. One of those options is the Twin Mill. This means that the number of favorable outcomes is $1$ and the number of possible outcomes is $8.$

$Number of favorable outcomes=1Number of possible outcomes=8 $

Now, write the ratio of the number of favorable outcomes to the number of possible outcomes to determine the probability of Davontay getting the Twin Mill. $P(Getting a Twin Mill)=81 ⇕P(Getting a Twin Mill)=0.125 $

The probability of Davontay getting a Twin Mill is $0.125$ or $12.5%.$
b A similar process as in Part A can be applied to find the probability of Kevin obtaining the book of his choice. Kevin has a selection of five books, one of which is

The Sparkle Tales. This yields one favorable outcome out of five possible outcomes.

$Number of favorable outcomes=1Number of possible outcomes=5 $

Calculate the ratio of the number of favorable outcomes to the total number of possible outcomes to determine the probability of Kevin getting the book he wants. $P(Getting The Sparkle Tales Book)=51 ⇕P(Getting The Sparkle Tales Book)=0.2 $

The probablity of Keving getting The Sparkle Tales books is $0.2$ or $20%.$
Discussion

Complementary events are pairs of events where only one can occur at a time. The complement of an event $A$ is written as $A_{′},$ $A_{c},$ or $A.$ It represents all possible outcomes that are not part of event $A.$ For instance, when rolling a fair six-sided die, rolling an even number and rolling an odd number are complementary events.

The probability of an event and its complement always adds up to $1.$

$P(Event)+P(Complement)=1 $

Example

Kevin was able to get the book he wanted, but Davontay, unfortunately, did not get the Twin Mill he had hoped for. To cheer Davontay up, Kevin took him to the ball pit and made a bet. He challenged Davontay to pick one ball from the pit blindly. If the ball he picks is not red, Kevin will give his extra Twin Mill that he has at home to Davontay.

Assume there are $240$ balls, equally divided into blue, red, yellow, green, pink, and orange balls. What is the probability of Davontay selecting a non-red ball? Answer in fraction form.{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord text\"><span class=\"mord Roboto-Regular\">Not<\/span><span class=\"mord\">\u00a0<\/span><span class=\"mord Roboto-Regular\">a<\/span><span class=\"mord\">\u00a0<\/span><span class=\"mord Roboto-Regular\">red<\/span><span class=\"mord\">\u00a0<\/span><span class=\"mord Roboto-Regular\">ball<\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["\\dfrac{200}{240}","\\dfrac{5}{6}"]}}

Consider that the sum of the probabilities of two complementary events equals $1.$

When choosing a ball from a pit of $240,$ the probability of not getting a red ball or getting a red ball are complementary events. Recall that the probability of an event and its complement always adds up to $1.$

$P(Not a red ball)+P(Red ball)=1 $

The $240$ balls are divided into $6$ different colors — blue, red, yellow, green, pink, and orange. Divide $240$ by $6$ to determine the number of balls for each color.
$6240 =40 $

There are $40$ balls of each color. Use this information to find the probability of picking a red ball from the pit. It is given by ratio of the number of red balls to the total number of balls.
$P(Red ball)=24040 ⇒P(Red ball)=61 $

Substitute the probability of getting a red ball into the expression of the sum of the probabilities of the complementary events. $P(Not a red ball)+61 =1 $

Think about what number must be added to $61 $ to get $1.$ This value is $65 .$
$P(Not a red ball)+61 =1⇓65 +61 =1 $

Therefore, the probability of not picking a red ball is $65 .$ Davontay has quite a good chance of getting the Twin Mill from Kevin!
Discussion

A uniform probability model is a probability model in which every outcome of an experiment is equally likely to happen. Therefore, if an experiment has $n$ different outcomes, each has a probability of $n1 .$ For example, consider rolling a fair six-sided die.

Each outcome of rolling a die has the same probability of occurring. Since there are $six$ different outcomes, each has a probability of $61 .$

Example

Davontay and Kevin were given four games to choose from, but they could only pick one. Winning the selected game will earn them a prize.

To boost their chances of winning, they should assess the probability of winning each game based on its unique requirements. The games in question are based on a uniform probability model. 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red card"},{"id":2,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.73046875em;vertical-align:-0.009765625em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Bold textbf\">C<\/span><\/span><span class=\"mord\">.<\/span><\/span><\/span><\/span> Landing on blue"},{"id":3,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7109375em;vertical-align:0em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Bold textbf\">D<\/span><\/span><span class=\"mord\">.<\/span><\/span><\/span><\/span> Picking the letter M"}],[{"id":0,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:2.00744em;vertical-align:-0.686em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span 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class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:2.00744em;vertical-align:-0.686em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">3<\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span 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class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span>"}]],"lockLeft":true,"lockRight":false},"formTextBefore":"","formTextAfter":"","answer":[[0,1,2,3],[0,1,2,3]]}

{"type":"choice","form":{"alts":["<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7109375em;vertical-align:0em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Bold textbf\">A<\/span><\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7109375em;vertical-align:0em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Bold textbf\">B<\/span><\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.73046875em;vertical-align:-0.009765625em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Bold textbf\">C<\/span><\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7109375em;vertical-align:0em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Bold textbf\">D<\/span><\/span><\/span><\/span><\/span>"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":1}

Find possible and favorable outcomes for each event. Consider that each outcome has an equal chance of occurring under the uniform probability model.

The probability of each event will be determined by analyzing all possible outcomes and their corresponding favorable outcomes. In addition, given that each experiment follows a uniform probability model, the probability of each outcome occurring is equal.

This experiment consists of picking a number from the set formed by the numbers from $1$ to $20.$ The event of interest is that the chosen number corresponds to a multiple of $4.$ Begin by highlighting the multiples of $4$ from the given set.

Out of the twenty numbers in the set, there are five multiples of $4.$ Therefore, the probability of selecting a multiple of $4$ is $5$ out of $20,$ which can be expressed as the following ratio.$P(Picking a multiple of4)=205 ⇕P(Picking a multiple of4)=41 $

A standard deck of cards consists of $52$ cards divided into four suits — spades, clubs, hearts, and diamonds. The colors of the suits are red for hearts and diamonds and black for spades and clubs.

Each suit in a deck of cards contains $13$ cards. Since there are two red suits, there are a total of $26$ red cards in the deck. Divide $26$ by the total number of cards in the deck, $52,$ to determine the probability of drawing a red card.$P(Drawing a red card)=5226 ⇕P(Drawing a red card)=21 $

The roulette wheel is divided into three equal sections representing the different possible outcomes. Of these three outcomes, only one is colored blue, representing the only favorable outcome.

Therefore, the probability of landing on the blue section is given by $31 .$

For this experiment, a letter will be randomly selected from the English Alphabet. The event is to get the letter M and every letter has an equal chance of being chosen.

Since there are $26$ letters in the alphabet and only one of those is the letter M, the probability of selecting the letter M is $261 .$

All games have been analyzed. The table below pairs each of them with their corresponding probability.

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 $4$ | $41 $ |

Randomly drawing a card from a standard deck of cards | Drawing a red card | $21 $ |

Spinning a $3-$colored roulette wheel evenly divided | Landing on blue | $31 $ |

Randomly selecting a letter from the English Alphabet | Picking the letter M | $261 $ |

Out of all the games available, the likelihood of drawing a red card from a standard deck of cards is higher. Therefore, if Davontay and Kevin want to increase their chances of winning, they should choose this game.

Discussion

When all outcomes in a sample space are equally likely to occur, the theoretical probability of an event is the ratio of the number of favorable outcomes to the number of possible outcomes.

$P(event)=Number of possible outcomesNumber of favorable outcomes $

The probability of $A$ can be found by using the given ratio.

$P(A)=Number of possible outcomesNumber of favorable outcomes =63 $

After simplification, it is obtained that $P(A)$ is equal to $21 .$ Consequently, there is a $50%$ chance an even number is rolled.$.$

Discussion

Experimental probability is the probability of an event occurring based on data collected from repeated *trials* of a probability experiment. For each trial, the outcome is noted. When all trials are performed, the experimental probability of an event is calculated by dividing the number of times the event occurs, or its frequency, by the number of trials.

$P(event)=Number of trialsNumber of times the event occurs ⇕P(event)=Number of trialsFrequency of the event $

The experimental probability of tails as an outcome is close to the theoretical probability of $0.5.$

Example

Kevin and Davontay spun the roulette wheel at a restaurant and were close to winning the Theme Park Passes, but ended up winning a Toy Shopping Spree. They decided to recreate the roulette wheel at home and conducted an experiment by spinning it $120$ times.
### Hint

### Solution

The experimental probability of landing on the Theme Park Passes can be determined by dividing the frequency of this event by the total number of spins.

The graph displayed below shows the results of their experiment.

a Find out the experimental probability of winning the Theme Park Passes using the experimental results gathered by Davontay and Kevin. Write it in fraction form.

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b When comparing the experimental probability obtained in the experiment to its theoretical probability, which statement accurately describes the relationship between the two probabilities?

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a Use the graph to determine the frequency of Theme Park Passes. Divide the frequency by the total number of spins to determine the experimental probability.

b Determine the theoretical probability of landing on Theme Park Passes. Write both probabilities side by side and compare them.

a The given graph shows how often the roulette wheel landed on different prize regions during spins carried out by Davontay and Kevin. The specific focus is on the Theme Park Passes section, which was a frequency of $18.$

$Experimental ProbabilityP(Theme Park Passes)=12018 ⇕P(Theme Park Passes)=203 $

Therefore, the experimental probability of landing on Theme Park passes is $203 .$
b Note that the wheel is divided into six equal sections, each representing a different prize. This means that the theoretical probability of landing on Theme Park Passes is $1$ out of $6.$

$Theoretical ProbabilityP(Theme Park Passes)=61 $

The experimental and theoretical probability of landing on Theme Park Passes can be rewritten into decimal numbers to compare them. This way, it can be determined which statement is correct. $P(Theme Park Passes)$ | |
---|---|

Experimental Probability | Theoretical Probability |

$203 =0.15$ | $61 =0.16$ |

It is reasonable to conclude that the experimental probability is close to the theoretical probability because $0.15$ is close to $0.16.$

Closure

This lesson discussed simple events and their probabilities. These are events that cannot be broken down further and only have one outcome. However, events can also be made up two or more simple events. To illustrate this, consider the example of flipping a fair coin and rolling a fair six-sided die.

In this experiment, multiple individual events or outcomes are combined to create compound events. Examples include getting heads (H) on a coin and rolling a $3$ on the die, or getting tails (T) on the coin and rolling a $6$ on the die.

Flipping a Coin and Rolling a Die | |||||
---|---|---|---|---|---|

Possible Outcomes | |||||

$(H,1)$ | $(H,2)$ | $(H,3)$ | $(H,4)$ | $(H,5)$ | $(H,6)$ |

$(T,1)$ | $(T,2)$ | $(T,3)$ | $(T,4)$ | $(T,5)$ | $(T,6)$ |

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