cadaVR anatomy by Mozaik Education

Certain event, impossible event

Nine frogs are sitting on the shore of a lake. Three of them jump into the water. How many yellow frogs are left on the shore?

Example 1

Two regular dice are thrown. The following outcomes are called events. Which of the following outcomes are certain, possible and impossible?

The sum of the numbers thrown with two dice is \latex{ 1 }.
The product of the numbers thrown with two dice is \latex{ 1 }.
The sum of the numbers thrown with two dice is a positive number not larger than \latex{ 12 }.
The product of the numbers thrown with two dice is a composite number.

Solution

We can throw numbers between \latex{ 1 } and \latex{ 6 } with both dice.

The smallest possible sum of the numbers is \latex{1 + 1 = 2} It is impossible that the sum of the two numbers is \latex{ 1 }.
If both numbers are \latex{ 1 }, their product is \latex{1\times 1=1}. In every other possible outcome, the product of the numbers on the dice is larger than \latex{ 1 }. This means that the event when the product of the numbers on the dice is \latex{ 1 } is possible but not certain.
The smallest possible number thrown with the dice is \latex{ 1 }, while the largest is \latex{ 6 }. The sum of the numbers is at least \latex{ 2 } and \latex{ 12 } at most. Therefore, the event when the sum of the numbers on the dice is a positive number not larger than \latex{ 12 } will certainly occur.
A composite number has factors other than \latex{ 1 } and itself. For example, if you throw \latex{ 2 } and \latex{ 3 }, their product is \latex{2\times 3=6}, which is a composite number. However, if you throw \latex{ 1 } and \latex{ 5 }, their product is \latex{1\times 5=5}, which is a prime number. If you throw \latex{ 1 } and \latex{ 1 }, their product is \latex{1\times 1=1}, which is not a composite nor a prime number. The event when the product of the numbers on the dice is a composite number is possible but not certain.

Example 2

\latex{ 9 } frogs are sitting on the lakeshore: \latex{ 3 } green, \latex{ 2 } yellow and \latex{ 4 } brown. Three of them jump into the water. Which of the following events will certainly occur?

None of the frogs remaining on the shore are green.
All the frogs remaining on the shore are green.
There is at least one brown frog remaining on the shore.

Solution

Event: None of the frogs remaining on the shore are green.

This occurs if three green frogs jump into the water. In every other case, at least \latex{ 1 } green frog remains on the shore. This outcome is possible but not certain.

Event: All the frogs remaining on the shore are green.

Since there are \latex{ 6 } frogs remaining on the shore and only \latex{ 3 } green frogs, it is impossible that every frog remaining on the shore is green.

Event: There is at least one brown frog remaining on the shore.

Since at most \latex{ 3 } of the \latex{ 4 } brown frogs jump into the water, there will certainly remain a brown frog on the shore.

is the only outcome that will certainly occur.

Example 3

There are \latex{ 14 } red, \latex{ 8 } yellow, and \latex{ 5 } white roses in a vase. At least how many do you have to take out to have certainly

two of the same colour;
one yellow rose?

Solution

If you take three roses out, it is possible that all of them are of different colours (but not certainly).

If you take a fourth one out, its colour will certainly match one of the previous ones, as there are only three colours.

So you must take out at least four roses to make sure you have two roses of the same colour.

When you take out \latex{ 19 } roses, it is possible that you remove \latex{ 14 } red and \latex{ 5 } white roses (there might be a yellow among them, but it is not certain).

When you take out \latex{ 20 } roses, there will certainly be a yellow among them because there are \latex{ 19 } red and white roses.

So you must take out at least \latex{ 20 } roses to make sure you have a yellow among them.

Probability Game

Several players can play the following game. Everybody has a figure. Take turns throwing a dice. The number you throw is the number on the upper face of the dice. You can leave the 'START' field if you throw a \latex{ 6 }. Afterwards, you can advance the same number of fields you throw with the dice.

If you step on a field with an arrow, move your figure to the field where the arrow points. If you step on a field that can be divided by \latex{ 9 } without leaving a remainder, you can throw again.

Every field that can be divided by \latex{ 7 } without leaving a remainder is a 'lucky field'. When you step on a lucky field, you can throw the dice three times. The first number shows which lucky card you get. Then, you roll the dice two more times. If the statement on the lucky card is valid for the throws, you can advance the sum of the last two rolls.

On field \latex{ 49 }, the player chooses the lucky card.

The player who reaches the finish first wins the game.

Lucky cards:

The sum of the numbers is \latex{ 7 }.
The sum of the numbers is even.
The sum of the numbers is a prime number.
The sum of the numbers is \latex{ 6 } or \latex{ 8 }.
The sum of the numbers is not smaller than \latex{ 13 }.
The sum of the numbers is not greater than \latex{ 12 }.

Questions

Is it possible to be on field \latex{ 14 } after \latex{ 22 } throws?
Is it possible to reach the finish in \latex{ 12 } steps?
What is the minimum number of steps required to reach the finish from field \latex{ 24 }?
At least how many steps do you need to get to the finish from the Start field?
Decide whether the following events are possible, impossible or certain.

You reach the finish with \latex{ 11 } throws from field \latex{ 44 }.
You reach the finish from field \latex{ 81 } with \latex{ 5 } throws at most.
You leave the START field after the first throw.
You leave the START field after the tenth throw.
It takes you \latex{ 3 } throws to reach the finish from field \latex{ 33 }.

Which lucky card should you choose on field \latex{ 49? }
Which lucky card reduces your chances of winning the most?
Do you have a better chance of winning with lucky card \latex{ 1 } or \latex{ 6 ?}

Exercises

{{exercise_number}}. There are \latex{ 6 } marbles in a hat: \latex{ 2 } red and \latex{ 4 } green. You take out \latex{ 3 } marbles. Which of the following events are certain, impossible, and possible but not certain?

The colour of the three marbles that you take out is the same.

All the marbles that you take out are red.

There is at least one green among the marbles you take out.

None of the marbles you take out is red.

Two of the marbles you take out have the same colour.

{{exercise_number}}. Your wallet contains the following banknotes: three \latex{ 5 }-\latex{ euro } banknotes, four \latex{ 10 }-\latex{ euro } banknotes, and two \latex{ 50 }-\latex{ euro } banknotes. (We do not know their order.) You take out four banknotes without looking into the wallet. Which of the following events are certain, impossible and possible but not certain?

You took out at least €\latex{ 25 }.

There are two identical banknotes among the ones you took out.

The banknotes you took out are all \latex{ 5 }-\latex{ euro } banknotes.

At most, you took out €\latex{ 120 }.

You took out exactly €\latex{55 }.

There are €\latex{10 } left in your wallet.

{{exercise_number}}. \latex{ 5 } freely chosen consecutive natural numbers are selected. Which of the following events are certain, impossible, and possible but not certain?

There is a number divisible by \latex{ 5 }.

Four of the numbers are even.

All of the numbers are odd.

There are two numbers divisible by \latex{ 4 }.

There are two numbers divisible by \latex{ 3 }.

{{exercise_number}}. At least how many consecutive natural numbers must be chosen to make sure that their product is divisible by

a) \latex{ 2; }

b) \latex{ 8; }

c) \latex{ 12; }

\latex{ 15 ?}

{{exercise_number}}. Every digit is written on a card and put into a hat. At least how many cards must be drawn to make sure that

there certainly is an even number among the cards;

there is a number divisible by \latex{ 3 } among them?

Quiz

At least how many students must go to the same school to make sure there are two students whose birthdays are on the same day?

Mathematics 6.

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