Statistic Math

Written Homework # 4

I have attempted this assignment honestly and the material submitted is my own work. I have only worked with the other group members listed on this page. I have not copied from any source, and cited any materials I have used outside of lecture notes and the textbook.

Names: Signatures:

Chapter Objectives (ch 13, 14, 15, 16): Construct Venn Diagrams from Contingency Tables

Use Venn Diagrams to compute basic and compound probabilities Describe compound events in words

Use Tree Diagrams to compute conditional probabilities Compute Expected Value of a game

Use Geometric and Binomial Distributions to compute probability

Due Date: Monday, July 24

Statistics Summer 2017

1 Below are contingency tables describing the demographics of the current US Congress. Our probability experiment is to choose one member of congress at random. Let A be the event that this person is Female. Let B be the event that this person is White. Let C be the event that this person is Republican.

. Females Males Totals:

White 50 287 337

Non-White 34 63 97

Totals 84 350 434

. Republican Democrat Totals:

White 227 110 337

Non-White 13 84 97

Totals 240 194 434

. Females Males Totals:

Republican 22 218 240

Democrat 62 132 194

Totals 84 350 434

(a) Construct a 3-circled Venn Diagram showing the demographics of the Congress. Label the circles A, B and C. You will need to use the fact that there are 19 Republicans who are White and Female. This number will go in the center of your Venn Diagram.

(b) For each of the following probabilities, describe in words what we are computing the probability of. Then compute the probability. Justify your answer.

i. P (A ∪ B)

ii. P (B|C)

iii. P (AC ∩ B)

iv. P ((B ∪ C)C )

v. P (C ∩ (A ∪ B)).

(c) Are any of these events (A, B, C) mutually exclusive? Why or why not?

(d) Show that each event A, B, C are dependent with the other two.

(e) Bonus: Find equivalent ways to write the last two probabilities. Try using Venn Diagrams, or consider how else you could describe the event.

2 A card game is played as follows: There are 5 players, each with 5 cards in hand numbered 1-5. Each player shuffles his or her deck. The first player turns over the top card in his or her deck. The second player turns over the top card in his or her deck. If the number matches, the second player wins. If not, the third player turns over the top card in his or her deck. If it matches either of the previous cards played, player 3 wins. If it does not match either, the fourth player turns over a card. If it matches any of the previous cards, player 4 wins. If not, player 5 turns over a card. If that card matches any of the previous cards played, player 5 wins. If not, the first player wins.

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Statistics Summer 2017

(a) Draw a tree diagram showing the probability of each player winning. Which player has the highest probability of winning?

(b) Suppose that it costs $4 to play. If someone wins before its your turn, you get $2 back. If someone wins after your turn (including player 1), you lose all $4. If you win, you get your initial $4 back and you get another $3. Compute the expected value for each player. How much money will each player win/lose if the game is played 100 times?

(c) Set the price and prizes so that the person most likely to win loses an average of 5 cents per game. Show that your pricing scheme does this.

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