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Here are a few recommended readings before getting started with this lesson.
A local $3-$on$-3$ beach volleyball tournament was hosted in South Beach. Heichi, Izabella, and Kevin pulled off a huge upset and won the tournament. The reward is a bicycle that flies as wind gusts propel the rider anywhere they so please to go. Everyone wants it, but, unfortunately, there is only one. Therefore, each player suggests a method to decide who gets to keep the highly coveted bicycle.
Considering the three suggestions, which method is fair?
Try to determine which game leads to a fair decision. In order to do this, complete the following steps.
A fair decision is a decision for which all events are equally likely to happen, meaning that the probabilities of the events are the same. Fair decisions can be made by using several methods, such as selecting names from a hat, flipping a coin, rolling a die, drawing a card from a standard deck of cards, using a random number generator, and so on.
Suppose that the captains of two baseball teams $A$ and $B,$ have to decide which team bats first. To do so, the captains will roll one die each, simultaneously.
One example of a fair method and one example of an unfair method will be presented for this situation.
The fairness of the mentioned methods will be explained one at a time.
This method suggests rolling two dice and calculating the sum of the numbers obtained. Since rolling each die has $6$ possible outcomes, the number of all possible combinations is $36.$
Recall that if the sum of the numbers is less than $7,$ then Team $A$ bats first. For a clearer understanding, highlight all the outcomes which satisfy this event.
After counting, there is a total of $15$ favorable outcomes. Using this information, the probability of Team A batting first can be calculated.$Favorable outcomes=15$, $Total outcomes=36$
$ba =b/3a/3 $
Calculate quotient
Round to $2$ decimal place(s)
Again, there are $15$ favorable outcomes for the event. Therefore, the probability that Team $B$ bats first is also about $0.42.$ Since two events, Team $A$ batting first and Team $B$ batting first, have equal probabilities, this method is fair.
This method suggests rolling two dice and multiplying the numbers obtained. Since rolling each die has $6$ possible outcomes, there are $36$ possible products.
According to this method, if the product of the numbers is even, then Team $A$ bats first. Hence, start by identifying the favorable outcomes which are the even numbers in the table.
There are $27$ favorable outcomes for Team $A$ out of $36$ possible outcomes. With this information, the probability of Team $A$ batting first can be calculated.$Favorable outcomes=27$, $Total outcomes=36$
$ba =b/9a/9 $
Calculate quotient
$Favorable outcomes=9$, $Total outcomes=36$
$ba =b/9a/9 $
Calculate quotient
In the last daring coin flip, Emily won and devoured the red velvet cake. The girls enjoyed the game so much that they decided to continue playing these games of probability just for fun. This time they will use a standard deck of cards.
Dominika will first remove all of the clubs and diamonds from the deck. Emily will then draw two cards in succession without replacing them. If she draws two cards with the same suit, then Emily wins. If she draws two face cards, then Dominika wins. For any other outcome, they draw again.
Try to determine if the game leads to a fair decision. To do so, complete the following steps.
Note that the event of drawing two hearts and the event of drawing two spades are mutually exclusive, therefore, the probability on the right-hand side equals the sum of the individual probabilities.
Since there are $26$ cards in total, the probability of getting a heart on the first draw is $2613 .$ If the first card is drawn is a heart, then only $12$ hearts remain in the deck, and because the cards are not being replaced there are $25$ remaining cards in total. Hence, the probability of drawing a second heart is $2512 .$Substitute values
$ba =b/13a/13 $
Multiply fractions
$ba =b/2a/2 $
Substitute values
Add fractions
Calculate quotient
Substitute values
$ba =b/2a/2 $
$ba =b/5a/5 $
Multiply fractions
Use a calculator
Round to $3$ decimal place(s)
There are opportunities to make fair decisions in different contextual situations by the use of simulations.
For example, Mark and Tadeo want to eat dinner at different restaurants and they are not able to come together on a decision. It so happens that they both really like basketball. They then came up with an idea to make a free throw to determine the person who will choose the restaurant.
A company with $m$ employees wants to award a prize to $n$ employee(s) at random — it has been a good year. Propose a fair way to choose the random winners for the given values of $m$ and $n.$
Method I: Assign each person a number, write the numbers on slips of paper, and then draw one of them out of a hat.
Method II: Assign each person a number from $1$ to $8$ and then roll a fair octahedron die.
Method III: Flipp three unbiased coins and assign each employee one of $8$ possible outcomes.
Method I: Assign each person a number, write the numbers on pieces of paper, and then draw two pieces out of a hat.
Method II: Eight spades ranging from $2$ to $9$ are taken from a deck. The numbers are assigned to employees and then two cards are drawn without being replaced.
$n=8$, $r=2$
Subtract term
Write as a product
Split into factors
Cross out common factors
Cancel out common factors
Multiply
$1a =a$
Start by assigning a number from $1$ to $246$ to each employee and then use a technology-based random number generator to select one number. As an example, a graphing calculator can be used. First, push the $MATH $ button. Then, scroll to the right to the PRB menu and choose the fifth option called randInt(.
The function randInt$(a,b,c)$ outputs $c$ integers in the range from $a$ to $b,$ inclusive. Since $5$ integers from $1$ to $246$ are required in this case, evaluate randInt$(1,246,5)$ in the calculator.
The random number generator provides each number from $1$ to $246$ an equal chance of being chosen, so each employee has an equal probability of winning the prize. Therefore, this method is fair.
A chess club of $4$ members decided gift a hand-carved wooden chess set to one of its members. The club wants each member to have a chance of winning this gift based on how many games they have won this week.
Members | Number of Games Won |
---|---|
Ali | $2$ |
Maya | $3$ |
Diego | $1$ |
Dylan | $0$ |
Sample Method: Assign integers from $1$ to $6$ to each member according to the number of games they won. Then roll a die to determine the winner.
Why it is Fair: The outcomes of rolling a die are equally likely. Additionally, the probability of winning the gift for each member is proportional to the number of games they have won.
Find a method of assigning outcomes so that the chance of winning is proportional to the number of games won.
Members | Games Won | Numbers Assigned |
---|---|---|
Ali | $2$ | $1and2$ |
Maya | $3$ | $3,4,and5$ |
Diego | $1$ | $6$ |
Dylan | $0$ | None |
Members | Games Won | Numbers Assigned | Probability of Winning the Chess Set |
---|---|---|---|
Ali | $2$ | $1and2$ | $62 ≈0.33$ |
Maya | $3$ | $3,4,and5$ | $63 =0.5$ |
Diego | $1$ | $6$ | $61 ≈0.17$ |
Dylan | $0$ | $None$ | $60 =0$ |
Notice that the probability of winning the gift for each member is proportional to the number of games they have won this week. Therefore, rolling a die can be considered as a fair method to decide who gets the gift.
Let $X$ be a random number between $0$ and $1$ produced by a random number generator. In the applet, a random number is represented by the point on the number line.
At the beginning of the lesson it has been said that the team of Heichi, Izabella, and Kevin won the local $3-$on$-3$ beach volleyball tournament and there is a bicycle in the prize package. All players want it, but there is only one. Therefore, each player proposed a method to decide who gets to keep the new bicycle.
Which player's method is fair?Begin by identifying all possible outcomes of the experiments in each method. Then, calculate the probabilities to determine the fair method(s).
Each method will be examined one at a time.
Heichi suggests rolling two dice and multiplying the numbers obtained. Since rolling each die has $6$ possible outcomes, there are $36$ possible products.
According to Heichi's method, if the product of the numbers is odd, then Izabella keeps the bicycle. Hence, the outcomes which are the odd numbers should be identified in the table.
There are $9$ such outcomes, which is the number of favorable outcomes for Izabella out of $36$ possible outcomes. With this information, the probability of Izabella getting the bicycle can be calculated.$Favorable Outcomes=9$, $Total Outcomes=36$
$ba =b/9a/9 $
Calculate quotient
$Favorable Outcomes=10$, $Total Outcomes=36$
$ba =b/2a/2 $
Calculate quotient
Round to $2$ decimal place(s)
$Favorable Outcomes=17$, $Total Outcomes=36$
Calculate quotient
Round to $2$ decimal place(s)
Izabella suggested writing each player's name $5$ times on slips of paper and putting all $15$ pieces in a bowl. Then one slip is randomly picked, which determines who gets the bicycle. To investigate if this method is fair, begin by calculating the probability of keeping the bicycle for each player, then analyze the results.
Favorable Outcomes | Total Outcomes | Probability | |
---|---|---|---|
Heichi wins | $5$ | $15$ | $155 ≈0.33$ |
Izabella wins | $5$ | $15$ | $155 ≈0.33$ |
Kevin wins | $5$ | $15$ | $155 ≈0.33$ |
It was obtained that each player is equally likely to get the bicycle. Therefore, Izabella's method is fair.
Kevin suggested flipping a coin two times. To analyze his method, make a table of all possible outcomes.
According to Kevin's method, if two heads show, then Heichi gets the bicycle. Note that there is only $1$ outcome that two heads show out of $4$ possible outcomes. Using this information, the probability that Heichi wins can be calculated.$Favorable Outcomes=1$, $Total Outcomes=4$
Calculate quotient
$Favorable Outcomes=2$, $Total Outcomes=4$
$ba =b/2a/2 $
Calculate quotient