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Minds on

MINDS ON

This is the discussion icon. Consider the Following

You are conducting a game that uses a standard deck of 52 cards.  

This is an image of each card in a standard deck of cards on a green background. The suits are in rows with the cards in order from Ace to King.

Choose one of the following distributions and outline the rules of the game.  

  • Binomial
  • Hypergeometric
  • Uniform

You do not have to think about points.  What will the player have to do with the cards? What are the outcomes?  

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Action.

ACTION

This is the ePortfolio icon. Criteria for Distribution Type

Part of the learning goal for this activity is know when to apply each method you have learned for creating a probability distribution.  You will need to come up with criteria to look for in situations so that you can choose. Throughout the activity, you can keep coming back to the following question.

What criteria are required when using the following distributions;

  • Binomial
  • Hypergeometric
  • Uniform

Come back to this question as you develop more clarity throughout the lesson and in the discussion.

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This is the ePortfolio icon. Example 1

A study reports that 5% of adults are afraid to be home alone at night.  

This is an image of a woman with hands covering her mouth looking frightened.

If 20 people are randomly selected, calculate the following:

  1. The probability that 1 person is afraid to be home alone at night.
  2. The probability at least 3 of them are afraid to be home alone at night.
  3. The expected number of people who are afraid to be home alone at night.

Save your work and a comparison (definition:What was the same? What was different? What did you do well? What do you still need to work on?)to the solution found below.

Solution
  1. P of 1 equals 20 C 1 times 0.05 to the 1 times 0.95 to the 19 equals 0.3774 equals 37.74 percent
     
  2. P of x greater than or equal to 3 equals 1 minus P of x less than or equal to 2 equals 1 minus 0.3585 minus 0.3774 minus 0.1887 equals 0.0754 equals 7.54 percent
     
  3. E of x equals 0.05 times 20 equals 1
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This is the ePortfolio icon. Example 2

You draw 5 cards from a standard deck of 52 cards.  

This is an image of 5 cards on a white background, they are a royal flush of hearts.

If X is a random variable that represents the number of hearts you get, calculate the following:

  1. The probability of 3 or more hearts
  2. The expected number of hearts.  Describe what this number means.  

Save your work and a comparison (definition:What was the same? What was different? What did you do well? What do you still need to work on?)to the solution found below.

Solution
  1. P of x greater than or equal to 3 equals P of 3 plus P of 4 plus P of 5 equals 0.0815 plus 0.0107 plus 0.0005 equals 9.28 percent
     
  2. E of X equals 13 times 5 divided by 52 equals 1.25
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This is the ePortfolio icon. Example 3

Out of any 30 car accidents, authorities are analysing the likelihood that different numbers were caused by texting and driving.  A study suggests that 1 in 4 accidents are caused by texting and driving.  

This is an image of a driver of a car texting while driving. The image is from the backseat and his eyes are seen in the mirror looking at the phone.

Calculate the following:

  1. The probability that 10 or more of the accidents are caused by texting and driving.
  2. The probability that between 8 and 10, including 8 and 10, accidents are caused by texting and driving.
  3. The expected number, of the 30 accidents, that are caused by texting and driving.

Save your work and a comparison (definition:What was the same? What was different? What did you do well? What do you still need to work on?)to the solution found below.

Solution

Using spreadsheet:

  1.  P of x greater or equal to 10 equals 0.1966 equals 19.66 percent.     
     
  2. P of x greater than or equal to 8 and less than or equal to 10 equals P of 8 plus P of 9 plus P of 10 equals 0.1593 plus 0.1298 plus 0.0909 equals 38.00 percent
     
  3. E of X equals 30 times 0.25 equals 7.5
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This is the ePortfolio icon. Example 4

In the game "Deal or No Deal," players are given the opportunity to pick 1 case out of 26 cases that have different dollar values in them.

During the game, the player must open cases to reveal the amount of money they contain.  When the case reveals a prize amount, the player and the "Banker" know that that value is not in the case they selected at the beginning.

In the image, the player has revealed that the selected prize could not be any of the shaded numbers.  The prize could be any of the unshaded prizes in either column.

This image is of a screenshot showing the prizes offered on the TV game show deal or no deal.

by spilgames

Based on the unshaded prizes in either column, the banker has offered a buyout of $59,639. Should the player take the buyout? Why or why not? Provide work to support your decision.

Save your work and a comparison (definition:What was the same? What was different? What did you do well? What do you still need to work on?)to the solution found below.

Solution
This is an image of the probability distribution with the expected value column of all the outcomes left on the board.

Each probability is 1/13 and the expected value is the average of the outcomes. Note: the Dealer's offer was 59639, exactly half of the expected value).

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This is the ePortfolio icon. Example 5

There are 15 females in your class of 25.  You form a group of 5 and count the number of females.  Calculate the following.

  1. The probability that 3 or more in the group are not female.
  2. The expected number of females in the group.

Save your work and a comparison (definition:What was the same? What was different? What did you do well? What do you still need to work on?)to the solution found below.

Solution
  1. P of x less than or equal to 2 equals P of 0 plus P of 1 plus P of 2 equals 0.0047 plus 0.0593 plus 0.2372 equals 30.12 percent
     
  2. E of X equals 15 times 5 divided by 25 equals 3
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Binomial vs. Hypergeometric

In the last two activities, you saw a study regarding sausages having meat that is not advertised on the package.  

The same numbers were used, but the situation was slightly different.  The Binomial situation (shown on the right below) sampled 20 sausages with the assumption that 20% contained meat not advertised.  The Hypergeometric situation sampled 20 sausages from a total of 100 in which a random 20 had meat not advertised.  Using similar numbers, the results were very similar as well as you can see below. 

 

This is the dropbox icon. Binomial vs Hypergeometric

  1. Looking at the Binomial and Hypergeometric situations, list and describe the similarities and differences in the following solutions.
     
  2. For each of the situations below, answer the following:
    1. Referring to your criteria for selecting a probability distribution, what distribution would be used?
    2. Change the scenario so that the other distribution would be used.  For example, if it is Binomial, change to Hypergeometric.

Scenario 1:  

Your manager at work is giving away tickets for the local Lacrosse game.  She has 10 tickets: 4 in the blue section and 6 in the Green section.  She gives you two at random.  What is the probability that they are both in the blue section?

Scenario 2:  

40% of the music you listen to is by a Canadian artist.  If you listen to 10 songs, what is the probability that 5 of them are by Canadian artists?

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