Represent probabilities as ratios, proportions, decimals between 0 and 1, and percentages between 0 and 100 and verify that the probabilities computed are reasonable; know that if P is the probability of an event, 1- P is the probability of an event not o

Know that probabilities range from 0 to 1, where 0 is impossible for an event to occur and 1 is 100% certain that an event will occur. All other probabilities can be written as a fraction, decimal and/or percent. Probabilities can be used to tell whether an event is; certain, very likely, equally likely, less likely and impossible to occur. The chart below shows how each probability can be labeled based on its likeliness. Certain 1/1, 1.0 or 100% Very likely/More likely Any fraction more than ½, .5, but less than 1. (51% to 99%) Equally likely ½, 0.5, or 50% Less likely Any fraction less than ½, .5, but more than O. (49% - 1%) Impossible 0/1, 0.0, or 0% Understand that theoretical probability is shown as a ratio with the numerator showing favorable outcomes, the possibility of a specific event; and the denominator showing the sample space, set of all possible outcomes together as one. Show any given probability as a fraction, decimal and/or percent. Check to see that a probability makes sense. Understand that to find the probability of an event not occurring, you would use 1 (a 100% probability) – P (the probability that it would occur), 1-P.

A jar contains the names of 28 students in Mrs. Brown’s class. She has 12 girls in her class and 16 boys. If she reaches into the jar without looking and pulls out one name, what is the probability as a percent that she will draw a girls’ name?

(about 43%)

(2)

Anita’s teacher is allowing her to choose one prize from the class prize box, however she will have to be blindfolded. The prize box has a yoyo, 3 homework passes, an art pad, a multi-colored pen and 3 journals. What is the probability that Anita will get one of the homework passes? Write your answer as a fraction, decimal and percent.

(1/3, 0.333, about 33%)

(3)

James’ teacher is allowing him to choose one prize from the class prize box, however he will have to be blindfolded. The prize box has a yoyo, 3 homework passes, an art pad, a multi-colored pen and 3 journals. What is the probability that James will get the art pad? Write your answer as a percent and tell how likely it is that James will get the art pad.

(about 11%, less likely)

(4)

Timmy is choosing randomly from a sack of lollipops. In the sack are 4 grape, 5 cherry, 2 mystery, and 1orange. What is the probability that Timmy will get a mystery pop? How likely is it that he will get this flavor? Explain why your answer is reasonable.

(Timmy only has 1 in 6 chances of getting a mystery pop. This is only about a 17% chance. It is unlikely that Timmy will get the mystery pop. This answer is reasonable because most of the lollipops are cherry and grape, in fact 9 out of 12 of them. Timmy is most likely to get one of these two flavors.)

(5)

Timmy is choosing randomly from a sack of lollipops. In the sack are 4 grape, 5 cherry, 2 mystery, and 1orange. What is the probability that Timmy will not get a mystery pop?

(Timmy has 5/6 of a chance that he will not get a mystery pop. That is, he has an 83% chance of getting any flavor but mystery..

Children need to understand the theoretical probability of an event occurring. Probabilities range from 0 to 1, where 0 is impossible for an event to occur and 1 is 100% certain that an event will occur. Probabilities can be used to tell whether an event is; certain, very likely, equally likely, less likely and impossible to occur. The chart below shows how each probability can be labeled based on its likeliness.

Certain

1/1 or 100%

Very likely/More likely

Any fraction more than ½, but less than 1. 51% to 99%

Equally likely

½ or 50%

Less likely

Any fraction less than ½, but more than O. 49% - 1%

Impossible

0/1 or 0%

Probabilities can be written as a fraction, a decimal or a percent. You may want to discuss how we hear probabilities in the media with weather forecasts, etc…

It is important that your child understand that when he/she finds theoretical probability, that he/she must always compare the number of favorable outcomes (probability being sought out) to the total number of possible outcomes (total possible combinations or choices). It may help to discuss the meaning of the word theoretical or even break it down into the base word, theory. Remind you child that a theory is an educated guess about what could happen. This is why when we look at theoretical probability, we don’t actually try the experiment, we just think about all possible outcomes that could happen if we did.

The probability of an event is shown a P(n), where n is replaced with the event that is being solved for. So, if we were to be solving for the chances of spinning red on a spinner, we would write P(red) or P(r). Theoretical probability can be shown as a fraction. The numerator (top number) of the fraction will always be the number of favorable outcomes. The denominator (bottom number) of the fraction will always be the number of possible outcomes. The theoretical probability for any event would be shown as; P(event) = number of favorable outcomes

number of all possible outcomes

The words would be replaced with numbers to solve for a specific probability. Here’s an example of how to use the formula above to find the theoretical probability of rolling a even number on a number cube 1-6.

Write the probability you are seeking in the parenthesis after the P.

P (even)

Count the possible favorable outcomes. The favorable outcomes for this problem are all even numbers on the cube. 2, 4, and 6 are even. Hence, there are 3 favorable outcomes of even numbers.

P(even) = 3__

Count all possible outcomes. There are 6 numbers on the cube, which means a total of 6 possible outcomes.

P(even) = 3__

6

If possible, write the fraction in simplest form by dividing both the numerator and denominator by the greatest common divisor (largest number that will divide into both evenly).

3/6 are both divisible by 3. So, the theoretical probability in simplest form is 1/3.

(2)

Most standardized tests require students to place fraction answers in simplest form, this holds true for probabilities written as fractions, as well. Remind your child to simplify fractions by dividing to put them in the lowest form. For example, if I subtract two fractions and get 7/14, I need to simplify. There are two quick checks you can do to determine if a fraction needs to be reduced. The first is to determine if the larger number of the fraction is divisible by the smaller number. For my fraction, 14 is divisible by 7 because 7x2=14. So, I simply divided both numbers by 7 and I get 77=1 (new numerator) and 147=2 (new denominator). The simplified fraction is ½. The second quick check is to determine if the numerator and denominator have a factor or factors in common. If they do, you choose the greatest common factor and divide both the numerator and denominator by it. So, if I have 6/8 and I know that 6 does not go into 8 evenly, I think about any other factors that both 6 and 8 are divisible by. I know that since they’re both even that they are divisible by 2. I divide each term by two and get ¾, my simplified fraction.

(3)

Review with your child how to change a fraction into a decimal. He/she will need to be able to do this for this skill. To find a probability we always first write it as a fraction. So, if there 3 red marbles, 4 blue marble and 5 green marbles, and we need to know P (red). We would first write our fraction as 3/12 (favorable outcomes/total possible outcomes). If we were asked to find the probability as a decimal, we would need to divide the numerator, 3, by the denominator, 12. Make sure that your child understands that the top number will always be divided by the bottom number, when changing a fraction into a decimal. So, if he/she is using a “division house”, the 3 would go inside and the 12 would be outside. 3 12 = 0.25. So, the probability would be represented as 3/12, ¼, or 0.25.

(4)

Review with your child how to change a fraction into a percent. He/she will need to be able to represent any given probability as a fraction, a decimal and/or a percent. In order to turn a fraction into a percent, you will need to do two things.

Follow the steps above to change the fraction into a decimal. (This step in simple terms is divide numerator by denominator). 3/12 = 3 12 = 0.25

Next, start at the decimal point and move it two places to the right. If the decimal stops behind the last number, you can just replace it with the percent sign. 0.25 = 25% However, if you have a decimal such as 0.375, you would keep the decimal and write 37.5%. Also, don’t forget, that if you have a decimal such as, 0.5 that you still need to move the decimal to places to the right. Since there is only one number to the right of the decimal, you add a 0 after the number. So, 0.5 is 50%.

(5)

Help your child understand why 1 – P shows the probability of a probability not happening. Begin by reminding your child that 1 would be 100% certain that an event would occur. Discuss what P stands for, make sure he/she knows it stands for a given probability. Below is an example you can use to walk your child through understanding the expression 1 – P.

Sample problem: Claudia has a bag of candy. She has 5 green, 2 red, and 1 orange candy in her bag. If she reached in the bag without looking, what would be the probability that she would not get green.

To solve this problem, we need to first find the probability that Claudia would pull out a green. P(green) = 5/8.

Since we the probability of green, we can use it to find the opposite, not green by subtracting from one whole chance, 100%, or more simply 1. 1 – 5/8 = P (not green).

However, in order to solve for this, you will need to write 1 in fraction form so that subtraction is possible. Remember, when we subtract fractions, their denominators must be the same. So 1 can be written as 8/8 for this problem, since 8pieces/8pieces = one whole. 8/8 – 5/8 = 3/8. The probability of not getting green is 3/8.

(6)

Don’t be afraid to estimate a decimal or a percent by rounding. For example, if you get a probability fraction of 8/28 and you need to convert your answer into a decimal and percent, you would get the decimal 0.2857142. This decimal can be rounded to the nearest tenth. The 8 is in the tenths place and it is followed by the 5. Any number that is followed by 5 or more can be rounded up. This means 8 will be turned into a 9. Our new decimal become 0.29. This would mean our percent would be about 29%. Remember, if you have to round down, you don’t make the number one less, you keep it and drop the back numbers. For example, if you had the decimal 0.23498 and wanted to round it to the nearest tenth, you would be either keeping the 3 or making it a 4. When you look at the number to the right of the 3 it is a 4. Numbers that are 4 or less tell us to round down. This means we will keep the three and drop all the back numbers. So, 0.23498 is 0.23 when rounded.

The letters A-E-I-O-U are in a box. What is the probability that you could pull out an O on the first try? Write your answer as a fraction, decimal and percent.

(1/5, 0.2, 20%)

(2)

The letters A-E-I-O-U are in a box. What is the probability that you could pull out an A or E on the first try? Write your answer as a fraction, decimal and percent.

(2/5, 0.4,40%)

(3)

The letters A-E-I-O-U are in a box. What is the probability that you will not pull out a U on the first try? Write your answer as a fraction, decimal and percent.

(4/5)

(4)

What formula can be used to express the probability of an event not occurring? Explain why it works.

(1 – P. This works because 1 represents the total chances of getting any probability. If the take away the actual likeliness of an event from 1, you will be left with the unlikliness of an event.)

(5)

Mr. Gomez has placed strips of paper with all the months of the year in jar. What is the probability that he will reach in the jar on the first time and pull out a month with the letter “a” in it? Write the probability as a percent. Explain how likely this event would be.

(50%, equally likely)

(6)

Mr. Gomez has placed strips of paper with all the months of the year in jar. What is the probability that he will reach in the jar on the first time and pull out a month with a capital letter J? Write the probability as a decimal. Explain how likely this event would be.

(0.17, less likely)

(7)

Mr. Gomez has placed strips of paper with all the months of the year in jar. What is the probability that he will reach in the jar on the first time and pull out a month with the letter u not in it? Write the probability as a fraction, decimal and percent.

(7/12, .583, about 58%)

(8)

Mr. Gomez has placed strips of paper with all the months of the year in jar. What is the probability that he will reach in the jar on the first time and pull out a month with the letter a or e in it? Write the probability as a fraction, decimal and percent. Explain why your answer is reasonable.

(1/12, 0.83333, about 8%. This is reasonable, because only one month does not have an a or an e, July. One month out of 12 will give you a very unlikely event and small percent.)

(9)

Your valentine gave you a box of chocolates. Inside there are 4 caramels, 2 coconut, 3 toffee, 6 fudge, and 1 truffle. All of the chocolates look the same. What is the probability as a fraction, decimal and percent that the first chocolate you bite will be fudge? Explain why your answer is reasonable.

(6/16,3/8, 0.375, 37.5%. There are a total of 16 chocolates, 6 of them are fudge. Half of 16 is 8. So, 6 will be somewhat less than half. 37.5% is a reasonable amount less than 50%.)

(10)

Your valentine gave you a box of chocolates. Inside there are 4 caramels, 2 coconut, 3 toffee, 6 fudge, and 1 truffle. All of the chocolates look the same. What is the probability as a fraction, decimal and percent that the first chocolate you bite will be almond? Explain why your answer is reasonable.

(0/16, 0.0, 0%. There are not almond chocolates, so it will be impossible to bite into one.)

(11)

Your valentine gave you a box of chocolates. Inside there are 4 caramels, 2 coconut, 3 toffee, 6 fudge, and 1 truffle. All of the chocolates look the same. What is the probability as a fraction, decimal and percent that the first chocolate you bite will be chocolate? Tell the likeliness of this event.

(16/16, 1.0, 100%, Certain)

(12)

A spinner is divided into 8 equal sections. Each section is colored. The sections are colored as follows: 1 red, 2 blue, 2 green, 3 yellow. What is the probability of spinning a red on the first try? Write your answer as a fraction and percent. How likely is this?

(1/8, 12.5%, less likely)

(13)

A spinner is divided into 8 equal sections. Each section is colored. The sections are colored as follows: 1 red, 2 blue, 2 green, 3 yellow. What is the probability of not spinning a red on the first try? Write your answer as a fraction and percent. How likely is this?

(7/8, 87.5%, more likely)

(14)

A spinner is divided into 8 equal sections. Each section is colored. The sections are colored as follows: 1 red, 2 blue, 2 green, 3 yellow. What is the probability of spinning a blue or green on the first try? Write your answer as a fraction and percent. How likely is this?

(4/8, 50%, equally likely)

(15)

A spinner is divided into 8 equal sections. Each section is colored. The sections are colored as follows: 1 red, 2 blue, 2 green, 3 yellow. What is the probability of not spinning a blue or green on the first try? Write your answer as a fraction and percent. How likely is this?

(4/8, 50%, equally likely)

(16)

A spinner is divided into 8 equal sections. Each section is colored. The sections are colored as follows: 1 red, 2 blue, 2 green, 3 yellow. What is the probability ofspinning a yellow, blue or green on the first try? Write your answer as a fraction and percent. How likely is this?

(7/8, 87.5%, more likely)

(17)

A spinner is divided into 8 equal sections. Each section is colored. The sections are colored as follows: 1 red, 2 blue, 2 green, 3 yellow. What is the probability ofspinning a red, yellow, blue or green on the first try? Write your answer as a percent. How likely is this?

(100%, certain)

(18)

A spinner is divided into 8 equal sections. Each section is colored. The sections are colored as follows: 1 red, 2 blue, 2 green, 3 yellow. What is the probability ofspinning an orange on the first try? Write your answer as a percent. How likely is this?

(0%, impossible)

(19)

When rolling a number cube, what is the probability of getting a number 2 or greater than 5? Write your answer as a percent.

(about 33%)

(20)

When rolling a number cube, what is the probability of getting the number 1 or a number that is greater than or equal to 2? Write your answer as a percent.

(100%)

(21)

When rolling a number cube, what is the probability of not rolling a number greater than 4? Write your answer as a percent.

(about 67%)

(22)

The letters C-A-L-I-F-O-R-N-I-A are on tiles in a bucket. Which letter(s) have a 20% chance of being chosen first?

(A and I)

(23)

The letters C-A-L-I-F-O-R-N-I-A are on tiles in a bucket. Which letter(s) have a 90% chance not of being chosen first?

(C, L, F, O, R, N)

(24)

A bag has 3 tennis balls, 5 ping pong balls, 4 marbles, and 2 racquetballs. If the coach reaches in to the bag, which ball has only about a 14% chance of being chosen?

(racquetball)

(25)

A bag has 3 tennis balls, 5 ping pong balls, 4 marbles, and 2 racquetballs. If the coach reaches in to the bag, which two balls together only have a 43% chance of not being chosen?