7th Test - Probabilities (again)

1) If a woman has 2 children, what is the probability that they are both girls? _____25%______  
    1/2 * 1/2 = 1/4    you multiply probabilities for independent events.

2) If I roll 2 dice, what is the probability that no 6's will show? _____25/36______
      5/6 * 5/6   Each dies has a 5/6 chance of not being a 6. Multiply independent events.

3) If I roll 2 dice, what is the probability that at least one 6 will show? ______11/36______
     Some noticed that problems 2 and 3 represent all possibilities when throwing 2 dice, so
     1 - 25/36 = 11/ 36

4) Two students miss their final exam because they were out late partying. The phony excuse they gave their professor was that they had a flat tire. The professor gave them a retest, and on that test, he asked them "which tire?". What are the odds they get that question correct? (its correct if they have the same answer)    _____1/4_______

   There are 4*4 = 16 total ways that they could answer, but there are only 4 ways that their answers could match.  So, 4/16 = 1/4

5) Suppose I draw 2 cards from a normal deck of playing cards and they are both aces. If I draw another card at random (without replacing the first 2 cards), what are the chances that I draw another ace? ____1/25____________

   2 aces left /  (52 - 2 aces which have been drawn already) = 1/25

6) a. Suppose I have 4 letters and 4 labels. Each label corresponds to one of the letters. If I apply the labels randomly, what are the chances that I get all 4 labels on their correct letters? ______1/24_________
 
   There are 24 total different ways for 4 labels to be attached to 4 envelopes.
   You can put the first label on one of 4 envelopes,
   the second on one of 3, and the 3rd on one of the 2 remaining envelopes.
   There is only one way to affix the last label to the last envelope.
    So we have 4*3*2*1 = 24 total variations, but only one way in which they are all correct.
 
b.   What are the chances that I get 3 of the 4 labels on their correct letters? _____0______
      I dare you to show me one way in which you can have exactly 3 labels correct.
      You cannot do it.
c.   What are the chances that I get 2 of the 4 labels on their correct letters? _____1/4_______
        There are 6 ways to get 2 labels correct
            1, 2 | 1,3 | 1, 4 | 2, 3 | 2, 4 | 3, 4   so 6/24 = 1/4
d. What are the chances that I get 1 of the 4 labels on their correct letters? ____1/3_____
       There are 8 ways to get just 1 label correct, so 8/24 = 1/3

7) In my sock drawer I have 7 red socks, 9 blue socks, and 5 green socks. If I pull out socks one at a time, what is the most socks I could withdraw and still not have a pair?  ____3____

   The worst you can do is draw a red, blue, and green sock, but after that you must get a pair.

8)  If I roll 2 dice, how many different ways can I roll a 7? ___6___
 
     1, 6 | 2, 5 | 3, 4 | 4, 3 | 5, 2 | 6, 1

9)  Suppose you are taking the STAR test and all 100 questions each have 5 possible answers. If you guess on all of them, what score do you expect to get?  _______20%________

   You expect to get  1/ 5 * 100 = 20 question correct or 20% of them.

10)  If you draw a single card from an ordinary deck, what are the chances that it is a 3 or a Jack? _______2/13__________

  Mutually exclusive events add. You cannot draw a card that is both a 3 and a Jack., so
    (4 + 4)/ 52 = 8/52 or 2/13