NEW EC-6 Math: Probability and Statistics

A fourth-grade teacher gives her class a math quiz worth 100 points directly after the lesson has been presented trying to determine the level of understanding before students start to do the assignment. The scores on the quiz are normally distributed with a mean of 50 and a standard deviation of 10. Approximately what percentage of the students failed the quiz with a score of less than 70?

. 97.5% In a normal distribution, approximately 95 % of all data fall within two standard deviations of the mean: either to the left or to the right. Taking ½ of 95 is 47.5. In a normal distribution, 50% of the data is below the mean. So, adding the 50% with the 47.5% (from the mean to two standard deviations to the right of the mean) gives 97.5%

Four stage framework that attempts to explain the learning process for probability:

1. Learners begin at the subjective level, at which they are easily swayed by personal experiences when making probabilistic statements. 2. Transitional learners begin to recognize the importance of organizing information. 3. Involves students becoming informal quantitative thinkers 4. students work at the purely numerical level in which they understand the nuances of numerical argument and use sophisticated procedures to determine numerical facts.

Students need to demonstrate an understanding of probability and statistics and be able to do the following:

1. use experimental and theoretical probability to make predictions 2. use statistical representations to analyze data

You roll a fair 6-sided die. What is the probability you roll an even number?

1/2 or (0.5) - There are 3 possible outcomes (2,4,6) -There are 6 possible outcomes 3/6 = 1/2 (0.5)

You're at a clothing store that dyes your clothes while you wait. You get to pick from 4 pieces of clothing (shirt, pants, socks, or hat) and 3 colors (purple, blue, or orange). If you randomly pick the piece of clothing and the color, what is the probability that you'll end up with socks that aren't blue?

1/4 x 2/3 = 2/12 = 1/6

A fair die has 6 faces numbered 1 through 6 that are each equally likely to show when the die is rolled. What is the theoretical probability that a fair die shows a 1?

1/6

If you flip three fair coins, what is the probability that you'll get all three heads?

1/8 1/2 x 1/2 x 1/2

Dave flipped a coin 20 times and got heads on 8 of the flips. Based on Dave's results, what is the experimental probability of the coin landing on heads?

2/5

You roll a fair 6-sided die. What is the probability (roll greater than 4)?

2/6 = 1/3 (or 0.33)

In a class of 10, there are 5 students who play soccer. If the teacher chooses 2 students, what is the probability that both of them play soccer?

2/9 We can think about this problem as the probability of 2 events happening. The first event is the teacher choosing one student who plays soccer. The second event is the teacher choosing another student who plays soccer, given that the teacher already chose someone who plays soccer. The probability that the teacher will choose someone who plays soccer is the number of students who play soccer divided by the total number of students: 5/10 Once the teacher's chosen one student, there are only 9 left. There's also one fewer student who plays soccer since the teacher isn't going to pick the same student twice. So, the probability that the teacher picks a second student who also plays soccer is 4/9 So, the probability of the teacher picking 222 students such that both of them play soccer is: 5/10 x 4/9 = 20/90 = 2/9

Quartiles are a measure of position often used to make comparisons especially when the volume of data is large. How many quartiles are used to divide a set of data?

3

You randomly draw marble out of a bag that contains 20 marbles. 12 of the marbles in the bag are blue. What is the probability of drawing a blue marble?

3/5 12/20 = 6/10 = 3/5

If you flip three fair coins, what is the probability that you'll get two tails and one head in any order?

3/8 Probability= Total possible outcomes ---------------------------- Favorable outcomes ​ If we flip three coins, there are 2 possible outcomes for each individual flip, so there are 2×2×2=8 8 total possible outcomes. Each outcome is equally likely. MAKE TABLE The probability of getting two tails and one head is 3/8

Dave continues flipping his coin until he has 100 total flips, and the coin shows heads on 47 of those flips. Based on these results, what is the experimental probability of the coin landing on heads?

47/100

If you flip a coin and roll a 6-sided die, what is the probability that you will flip a tails and roll at least a 2?

5/12 - The probability of getting a tails is 1/2 - The probability of rolling at least a 2 is 5/6, b/c there is 2, 3, 4, 5, 6, as possibilities So, 1/2 x 5/6 = 5/12

You roll a fair 6-sided die. What is the probability of not rolling a 5?

5/6

Omar ordered his sister a birthday card from a company that randomly selects a card from their inventory. The company has 21 total cards in inventory. 14of those cards are birthday cards. What is the probability (not getting a birthday card)?

7/21 = 1/3

Giovanna owns a farm. She is going to randomly select one animal to present at the state fair. She has 6 pigs, 7 chickens, and 10 cows. What is the probability for picking a chicken?

7/23

Table

A systematic or orderly list of values, usually in rows and columns

Permutations

All possible arrangements of a given number of items in which the order of the items make a difference, for example the different ways that a set of four books can be placed on a shelf.

Tosha has 8 coins in her pocket. She has a mixture of pennies, nickels, dimes and quarters, but she has no more than 3 of any coin. What is the largest amount of money she could possibly have? A. $1.11 B. $1.07 C. $1.21 D. $1.23

B. $1.07 - To satisfy the prompt given, there must be a minimum of $.08 in pennies and a maximum of $2.00 in quarters. - She will need to have the most number of the coins with the greatest value: quarters and dimes. So, that would be 3 quarters, 3 dimes, 1 nickel, and 1 penny. This totals $1.11.

Graphs

Generally used in grades PK to 6th grade - Pictorial, bar, pie, and line.

Histograms

Histograms are statistical graphs with vertical bars representing the frequency distributions of data.

What's the probability of rolling a one or a six?

P ( 1 or 6) = 2/6 = 1/3

Sets of data can be described by its-

Range: Difference between the greatest and the least numbers in the data set. Subtract these to find difference. Mean: Add all values and divide by number of values in a set. Median: middle value of all numbers. Oder values from least to greatest .If there are 2 values in the middle, find the number between two values by adding them together and dividing their sumb by 2 (mean) Mode: Value that appears the most in the set of data. If all the values in a set of data appear the same number of times, the set DOES NOT have a mode. It is possible to have MORE than 1 mode.

Ellen has a bag with 3 red marbles and 2 blue marbles in it. She is going to randomly draw a marble from the bag 300 times, putting the marble back in the bag after each draw. What is the best prediction for the number of times that Ellen will draw a blue marble?

The best prediction comes from multiplying the total number of trials by the probability of Ellen getting a blue marble on an individual draw. Best prediction= Number of trials x P(blue marble) Number of trials = 300 P(blue marble) = 2/5 300 x 2/5 We should not predict exactly how many times Ellen will draw a blue marble because drawing a marble is random. It is impossible to make exact predictions for outcomes of random events. The best prediction is that Ellen will draw a blue marble close to 120 times but probably not exactly 120 times.

Pictorial Graph

The most concrete representation of information. They represent a transition from the real object graphs to symbolic graphs. Used to introduce children in PK-1 to graphing.

Percentiles are commonly used to describe the measure of position rather than the measures of central tendency such as in standardize test scores. A student makes 530 on the Scholastic Aptitude Test (SAT) and the score is reported to be in the 79th percentile. Determine the choice that best describes what this means.

This student scored better than 79% of all the others taking the test.

Line Graphs

This type of representation is more abstract for children and is therefor more challenging for them. It tracks one or more subjects. One element is usually a time period over which the other element increases, decreases, or remains static.

Pie Charts

Used to help visualize relationships based on percentages of a subject. Parts of a whole.

Bar Graphs

Used to represent two elements of a single subject. e.g. Number of books read by a group of students.

Sample space

is the set of all possible outcomes of an experiment EX: A coin is flipped, it will land either heads or tails. Students learn to list the sample space, or all the possible outcomes: heads or tails. Sample spaces may also be listed on a chart or tree diagram.

Line Plot

represents a set of data by showing how often a piece of data appears in that set.