Section 1.1

 

4.   Suppose I bet on red at roulette and you bet on black, both bets on the same spin of the wheel.

 

      a.   What is the probability that we both lose?

      b.   What is the probability that at least one of us wins?

      c.   What is the probability that at least one of us loses?

 

Section 1.3

 

2.   Write down the expression in set notation corresponding to each of the following events.

 

      a.   the event which occurs if exactly one of the events A and B occurs;

      b.   the event which occurs if none of the events A, B or C occurs;

      c.   the events obtained by replacing "none" in part b by "exactly one", "exactly two", and "three".

 

4.   Let S={0,1,2} be the outcome space in a model for tossing a coin twice and observing the total number of heads.  Say if the following events can be represented as subsets of S.  If you say "yes", provide the subset; if you say "no", explain why.

 

      a.   the coin does not land heads both times;

      b.   on one of the tosses the coin lands heads, and on the other toss it lands tails;

      c.   on the first toss the coin lands heads, and on the second toss it lands tails;

      d.   the coin lands heads at least once.

 

5.   Think of the set S={HHH,HHT,HTH,HTT,THH,THT,TTH,TTT} as the outcome space for three tosses of a coin.  For example, the subset {HHH,TTT} corresponds to the event that all three tosses land the same way.  Give similar verbal descriptions for the events described by each of the following subsets of S.

 

      a.      {HHH,HHT,HTH,HTT}              

      b.      {HTH,HTT,TTT,TTH}

      c.      {HTT,HTH,HHT,HHH}

      d.      {HHH,HHT,HTH,THH}

      e.      {THT,HTT,TTH}                  

      f.      {HHT,HHH,TTH,TTT}

 

8.   Let A and B be events such that P(A)=0.6, P(B)=0.4, and P(AB)=0.2.  Find the probabilities of:

 

      a.   A union B   

      b.   Ac       

      c.   Bc        

      d.   AcB     

      e.   A union Bc

      f.    Ac Bc 

 

Section 1.4

 

2.   A light bulb company has factories in two cities.  The factory in city A produces two-thirds of the company's light bulbs.  The remainder are produced in city B, and of these, 1% are defective.  Among all bulbs manufactured by the company, what proportion are not defective and made in city B?

 

3.   Suppose:

      P(rain today)=40%; P(rain tomorrow)=50%; P(rain today and tomorrow)=30%.  Given that it rains today, what is the chance that it will rain tomorrow?

 

4.   Two independent events have probability 0.1 and 0.3.  What is the probability that

 

      a.   neither of the events occurs?

      b.   at least one of the events occurs?

      c.   exactly one of the events occurs?