Sample Spaces, Tree Diagrams, and the Counting Principle

Arabella is putting together a diversified stock portfolio by choosing 10 stocks from different sectors of the S&P 500.

● The S&P 500 has 76 companies in the Technology sector and she is going to choose 2

● The S&P 500 has 64 companies in the Healthcare sector and she is going to choose 3

● The S&P 500 has 31 companies in the Real Estate sector and she is going to choose 5

Calculate the total number of outcomes for each sector. Match each calculation with its description.

• possible Portfolios without restrictions

• ₇₆C₂ = 2,850 outcomes

• Real Estate sector

• ₆₄C₃ = 41,6664 outcomes

• Technology sector

• ₃₁C₅ = 169,911 outcomes

• possible Diversified Portfolios

• (2850)(41664)(169911) =

  20,175,639,926,400 different portfolios

• Healthcare sector

• ₅₀₀C₁₀ =

  245,810,599,810,000,000 different portfolios

Arabella is putting together a diversified stock portfolio by choosing stocks from different sectors of the S&P 500. ● The S&P 500 has 76 companies in the Technology sector and she is going to choose 2 ● The S&P 500 has 64 companies in the Healthcare sector and she is going to choose 3 ● The S&P 500 has 31 companies in the Real Estate sector and she is going to choose 5 You may have noticed that the Real Estate sector has less than half as many companies but produced more combinations than the other two sectors. Use the combinations formula to mathematically explain why 31C5 would produce more combinations than 76C2.

• The combinations formula favors larger groups when choosing a smaller subset because larger numbers in the numerator than the denominator will always yield a larger quotient.

• ₃₁C₅ results in a larger number because the formula calculates a higher value for smaller groups choosing larger subsets because dividing by larger factorials in the denominator will cancel out most of the multiplication in the factorial in the numerator.

• ₇₆C₂ is actually larger but was miscalculated in the question because multiplying 5 numbers that are about half as large should have resulted in a larger overall number.

• The combinations formula is not applicable in comparing these two scenarios.

How many different ways can you arrange 7 students in a row of desks?

• 5040

• 720

• 120

• 40320

• 840

A diner has a breakfast special. A customer can choose scrambled, fried, or poached eggs. The breakfast comes with a side of bacon, sausage, or fruit salad. The customer can choose coffee, tea, or milk. You make a sample space of all the possible combinations. How many different combinations of eggs, side, and drink does a customer have to choose from?

• 9

• 12

• 27

• 135

Tags

TEKS.MATH.7.6A

Which method of finding the total outcomes of an event allows you to see all possible outcomes?

• fundamental counting principle

• experimental probability

• theoretical probability

• tree diagram

Tags

TEKS.MATH.7.6A

The tree diagram shows the outcomes of rolling a die and flipping a coin.  
How many total outcomes are there?  

• 6

• 12

• 24

• 36

Tags

TEKS.MATH.7.6A

How many total outfit options are represented?

• 12

• 22

• 3

Tags

TEKS.MATH.7.6A

At a pizza shop you can choose thick or thin crust, red or white sauce, and toppings of pepperoni, cheese, or vegetarian.  How many different combinations are possible for someone who does not care for meat or white sauce?

• 3

• 24

• 9

• 4

Tags

TEKS.MATH.7.6A

The cafeteria is serving three kinds of sandwiches:  tuna (T), chicken, (C), and peanut butter (P).  They are also serving a choice of two drinks:  milk (M) or water (W).  Which shows the CORRECT Sample Space List of possible combinations?

• TW, CW, PW

• TW, TM, TC, CW, PW

• TW, TM, CW, CM, PW, PM

• TCP, MW

Tags

TEKS.MATH.7.6A

What is sample space?

• All of the Outcomes

• What Will Happen

• What Can Happen

• What I want to Happen

Tags

TEKS.MATH.7.6A