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  1. (5 points)  Use the binomial calculator to do this problem.


http://onlinestatbook.com/2/calculators/binomial_dist.html

Hopefully by April 2017, I will have 14 descendants – 13 boys and 1 girl.

The binomial equation is 1 = (p + q)n.

Let p be the probability of having a boy and q be the probability of having a girl.

  1. What is the value of p? (p = π in the binomial calculator.)

  2. What is the value of n that should be used?

  3. What is the probability of having 13 boys and 1 girl by chance? (include a screen shot of your answer)

  4. What is the probability of being “normal” (for people with 14 descendants) by having exactly 7 boys and 7 girls? (include a screen shot of your answer)

  5. (5 points) Last year McDonalds ran a promotion where it gave out Monopoly stickers with each purchase. If you got a winning sticker, you would get a small extra item free. The average person makes 3 purchases per visit, i.e. a Big Mac, French fries, and a drink; or an egg McMuffin, hash browns, and coffee; etc. Use the binomial calculator to do this problem.

  6. How many purchases does the average person make in 2 visits?

  7. What is the value of n that should be used?

  8. Let the probability of a single sticker being a winner be 10%. What is π?

  9. What is the probability of winning at least 1 prize (i.e. 1 or more prizes up to 6) after purchasing 6 items? (include a screen shot of your answer)

  10. What is the probability of winning exactly 1 prize after purchasing 6 items? (include a screen shot of your answer)

  11. What is the probability of winning 0 prizes after purchasing 6 items? (include a screen shot of your answer)

  12. (5 points) Calculate the odds of winning the Power Ball 2nd prize of $1 million. In order to win, you must match the numbers of 5 white balls numbered 1 to 69, and NOT match the red Power ball numbered 1 to 26. Write your final answer in 1/p instead of the fraction p. For example, ¼ = 4. 0.10 = 10.

  13. What is the probability of picking 1 of the 5 winning white balls from the bag?

  14. What is the probability of picking 1 of the remaining 4 winning white balls from the bag?

  15. What is the probability of picking 1 of the remaining 3 winning white balls from the bag?

  16. What is the probability of picking 1 of the remaining 2 winning white balls from the bag?

  17. What is the probability of picking the last remaining winning white ball from the bag?

  18. What is the probability of NOT picking the winning red Power ball from another bag? (Note: The white balls and the red Power balls are in different bags. They are not in the same bag.)

  19. Using the multiplication rule, what is the probability of winning the Power ball 2nd prize in 1/p? (Note: The Power ball ticket says the odds of winning are 1 out of 11,688,053.52. If you do not get something close to this number, then you did something wrong!)

  20. (10 points) The probability that a person has a certain fatal disease is 2% of the population. There is a test for detecting this disease. If you have the disease, the test gives the correct result 97% of the time. If you do not have the disease, then the test gives the correct result 95% of the time.


Fill in the following tree diagram.

  1. What is the probability that a person who gets positive test results actually has the disease? (true positive)

  2. What is the probability that a person who gets positive test results does not have the disease? (false positive)

  3. Of all of the people who test positive for the disease, what is the probability that they actually have the disease? P(true positive| positive test results)

  4. You are the medical advisor to the president. Based on the above statistics, would you recommend that everyone in the country get tested for this fatal disease?

  5. (10 points) Determine the poker odds of drawing the following hands from a standard 52 card deck.


What are the odds of drawing a royal flush? (Drawing AKQJ10 of the same suit.)

  1. What is the probability of drawing the A of spades for your first card?

  2. What is the probability of drawing the K of spades for your 2nd card?

  3. What is the probability of drawing the Q of spades for your 3rd card?

  4. What is the probability of drawing the J of spades for your 4th card?

  5. What is the probability of drawing the 10 of spades for your 5th card?

  6. Now there are 4 suits (spades, hearts, diamonds, and clubs). What number do you need to multiply the above probability to take into account that there are 4 suits?

  7. Now we drew the cards in a specific order (AKQJ10). But the order of the drawing of the cards does not matter in the end. What number do you need to multiply the above to get the final probability which takes into account that drawing order does not matter? (How many different ways can you arrange 5 different distinct objects?)


e.g. KQJ10A, QJ10AK, etc.

  1. Multiply the above numbers to get the final probability. Express you answer as 1/p (i.e., .10 = 10, .5 = 2)


(Note: The odds for getting a flush are 1/p = 649,740. If you did not get this number, then you did something wrong.)

  1. (10 points) Use the following values to fill in the Venn diagram.


A = “2” + “3”,    B = “3” + “4”

P(A) = 0.75, P(B) = 0.60, area(“3”) = 0.40

1 = area(“1”) + area(“2”) + area(“3”) + area(“4”)

  1. What is the value of area(“2”)?

  2. What is the value of area(“4”)?

  3. What is the value of area(“1”)?

  4. What is the value of P(A ∩ B)?

  5. What is the value of P(A Ụ B)?

  6. Are events A and B “mutually exclusive” in the mathematical sense? (yes or no, and explain your answer)

  7. Are the events A and B “independent” in the mathematical sense? (yes or no, and explain your answer)

  8. What is the value of P(A|B)?

  9. What is the value of P(B|A)?

  10. (10 points) Fill in the following table when using 2 dice.


 


































































1st roll/2nd roll123456
1
2
3
4
5
6


  1. Let event A be that the 1st die roll is an odd number. Put an A in the appropriate boxes.

  2. Let event B be that the 2nd die roll is an even number. Put a B in the appropriate boxes.

  3. What is P(A)?

  4. What is P(B)?

  5. What is P(A ∩ B)?

  6. What is P(sum of the 2 dice = 7|given A and B)?

  7. What is P(sum of the 2 dice = even number|given A and B)?

  8. Are events A and B mutually exclusive? (yes or no) Be sure to give your reasoning.

  9. Are events A and B independent? (yes or no) Be sure to give your reasoning.


 

  1. (10 points) Fill in the following contingency table to answer this question about the student population at a small liberal arts college.





























Arts majorScience majorTotal
Men75200
Women
Total350300


  1. If a student is chosen at random, what is the probability that it will be a man, P(M)?

  2. If an arts major is chosen at random, what is the probability that it will be a woman, P(W|A)?

  3. If a student is chosen at random, what is the probability that it will be a woman majoring in the sciences, P(W ∩ S)?

  4. If a student is chosen at random, what is the probability that it will be an arts major, P(A)?

  5. If a student is chosen at random, what is the probability that it will be a woman majoring in the arts OR a man majoring in the sciences, P[(W∩A) Ụ (M∩S)]?

  6. (10 points) You are planning on going to the National Harbor casino to play roulette. The roulette odds chart tells you how much money you will win if the ball lands on one of the numbers you bet on. For example, if you bet $1 on number 17 (a) and the ball lands on number 17, you will win $35 and get back your $1 bet as well. If you bet on 24 numbers (h) and the ball lands on one of the 24 numbers, you will get back $0.50 = (1/2 * $1) plus not lose your $1 bet. If the ball does not land on any of the numbers you bet on, then you just lose your $1 bet.


Roulette Payoff Odds Chart

  1. 35-to-1 : One number

  2. 17-to-1 : Two numbers

  3. 11-to-1 : Three numbers

  4. 8-to-1 : Four numbers

  5. 6-to-1 : Five numbers (including 0 and 00)

  6. 5-to-1 : Six numbers

  7. 2-to-1 : Twelve numbers (1 column, 1 to 12, 13 to 24, 25 to 36)

  8. ½-to-1 : Twenty-four numbers (two columns of 12 squares)

  9. 1-to-1 : Red (18 numbers)

  10. 1-to-1: Black (18 numbers)

  11. 1-to-1 : High (19 to 36)

  12. 1-to-1 : Low (1 to 18)

  13. 1-to-1 : Odd (18 numbers)

  14. 1-to-1 : Even (excluding 0 and 00 for 18 numbers total)


The expected pay off value for roulette is

E = Payoff odds*bet* P(win) - bet*P(lose)

Let bet = $10.

Choose two different payoff odds from the above table and calculate what the expected payoff value is for each of your choices. (Hint: The expected payoff value has to be negative so that the house can stay in business over the long term!)

For your two choices of payoff odds, are your two expected payoff values the same (yes or no)? (Hint: It should not matter which numbers you play. Every choice should lose the same amount of money. Otherwise, no one would play the numbers where the losses are higher.)