Homework #3 -- Suggested Answers

 

ISBE 3.4, 3.7, 3.10, 3.12, 3.15, 3.20, 3.21, 3.22, 3.33, 3.34

 

 

3.4

  1. 36%
  2. 47%
  3. 75%
  4. 25%

 

3.7

  1. 25.45%
  2. 9.09%

 

3.10

  1. 14%
  2. 86%
  3. 4%

 

3.12

  1. 40%
  2. 60%
  3. 55%
  4. 78%
  5. 17%
  6. 42.5%, 57.5%

 

3.15

  1. 7.2%
  2. The conditional probability of unemployment given the person is a male = 6.99%
  3. The conditional probability of unemployment given the person is a female = 7.47%

 

3.20

  1. Pr(F|E)= Pr(total is 7 | first is 5)=1/6=16.7%, i.e. the probability that the second roll is a 2. Pr(F)=6/36 (6 of the 36 possible rolls yield a total of 7), hence Pr(F|E) = Pr(F).
  2. Pr(G|E)= Pr(total is 10 | first is 5)=1/6=16.7%, i.e. the probability that the second roll is a 5. Pr(G)=3/36 (3 of the 36 possible rolls yield a total of 10), hence Pr(G|E) does not equal Pr(G).
  3. When betting on a ten from a roll of two dice, it does change the odds in your favor if you know the first die is a 5. However, when betting on a total of 7, a peek doesn't help.

 

3.21

  1. 26%
  2. 26% = (7.8/30)
  3. Yes, if Pr(F/E) = Pr(F) (26%=26%), then being in favor of the legalization of marijuana is statistically independent of living in the East.

 

3.22

a. i. Yes, whenever F is independent of E, then E must be independent of F.

ii. Yes, if F was independent of E, then E will be independent of F bar (the complement of F).

b. i. Pr(F|E) = 26%, Pr(F) = 26%

ii. Pr(F bar and E) = 22.2, Pr(E)= 30, thus Pr(F bar |E) = 74% and equals Pr(F bar) = 22.2% + 51.8% or 74%.

 

3.33

  1. Yes
  2. Yes
  3. Yes

 

3.34

  1. Chance = 1/4, odds = 1/3
  2. Chance = 1/2, odds = 1/1
  3. Chance = 1/13, odds = 1/12
  4. Chance = 4/13, odds = 4/9
  5. Chance = 3/51, odds = 3/49
  6. Chance = 24/51, odds = 24/27. (It can be higher, lower or the same. In part e we saw that the probability it is the same is 3/51, hence the probability that it is higher or lower is 48/51. There are as many ways of drawing 2 cards the first lower than the second as there are ways of drawing 2 cards the first higher than the second, therefore the probability that the card is higher than the next is half of 48/51 or 24/51. This calculation can also be done by conditioning on the value of the first card.)