1. College Students have different behaviors when it comes to wearing seat belts. Let N = the event that a college student never wears a seat belt.
A. Interpret the following in two seperate ways: P(N) = 0.0261.
1) The probability that a randomly selected student does not wear his or seat belt is 0.0261.
2) 2.61% of college students do not wear their seat belts.
B. Suppose you randomly select a college student. Would it be UNUSUAL to select one who never wear a seat belt? (see notes for "unusual" criteria)
          According to our criteria an event is unusual if its probability is less than 5%. So we say this event is unusual.
          2.
Guy has 0.4 probability of getting into Harvard.
Guy has 0.5 probability of getting into Yale.
Guy has 0.2 probability of getting into both Harvard and Yale.

A. Draw a Venn Diagram to Display the probabilities
B. For each question below, rewrite the probability question using the appropriate notation, show the appropriate definition and solve for the answer showing work.
(i) What is the probability that Guy does not get into Harvard?
P(not H) = 1 - P(H) = 1 - 0.4 = 0.6
(ii) What is the probability that Guy gets into either Harvard or Yale?
P(H or Y) = P(H) + P(Y) - P(H and Y) = 0.4+0.5-0.2= 0.7
(iii) What is the probability that Guy gets into Harvard, but not Yale?
P(H and not Y) = P(H) - P(H and Y) = 0.4-0.2=0.2
(iv) What is the probability that Guy does not get into Harvard and does not get into Yale? Make this the compliment of an "OR" statement. P(not H and not Y) = Pnot( H or Y) = 1 -P(H or Y) = 1 - 0.7 = 0.3
(v) Show mathematically why the events: "getting into Harvard" and "getting into Yale" are or are not mutually exclusive for Guy. P(H and Y) = 0.2 not = 0.

3.  The most common new car bought is Silver or White.  Here is a probability distribution table for the color choice of new cars.  Assume a car choice is not a mix of colors (hence, car-colors are mutually exclusive)

A. What type of probabilities are listed above? How were these probabilities obtained?
The above are experimental probabilities as they are relative frequecies of sample data.
B. What is the probability that the car chosen is a color other than the ones listed? 
P(other) = 1 - 0.431 = 0.539
C. What is the probability that a randomly selected vehicle is either silver or white? 
P(S or W) = P(S) + P(W) = 0.176+0.172=0.348
(for questions B through H, rewrite the question as a probability statement, show anappropriate definition, and show work leading to your solution)
D. If two cars are randomly selected, what is the probability that they are both white (assume independence)?
P(W1 and W2) = P(W1)P(W2)= (0.172)(0.172)=(0.172)^2
E. If three cars are randomly selected, what is the probability that they are all white (assume independence)?
P(W1 and W2 and W3) = P(W1)P(W2)P(W3)= (0.172)(0.172)(0.172)=(0.172)^3
F. If seven cars are randomly selected, what is the probability that they are all white (assume independence)?
P(all W) = (0.172)^7
G. If seven cars are randomly selected, what is the probability that at least one is white (assume independence)?
P(at least one W) = 1 - P(all not W) = 1 - (0.828)^7
H. If seven cars are randomly selected, what is the probability that at least one is not Black (assume independence)?
P(at least one not B) = 1 - P(all B) = 1 - (0.113)^7

4.  Provide a written description of the compliment (opposite) of the given event.
a.  When 10 students are tested for blood group, at least one of them has Group A blood.
None of them has Group A blood.
b.  When an IRS agent selects 12 income tax returns and audits them, none of the returns are found to be correct.
At least one is found to correct.
c.  I got all A’s in my classes.
I got a grade other than A in at least one of my classes.

5.  If you run a traffic light at an intersection equipped with a camera monitor, there is 0.15 probability that you will be given a traffic violation.  If you run a red traffic light at this intersection 5 times, what is the probability of getting at least one traffic violation (assume independence)?
Let T = given traffic violation : P(at least one T) = 1 - P(all not T) = 1 - (0.85)^5 = 0.9999

6.  A student misses many classes because of a malfunctioning alarm clock.  Instead of buying one new alarm clock, he decides to buy and use three.  What is the probability that at least one of his alarm clocks works correctly if each individual clock has a 99% chance of working correctly?  Does the student really gain much by using three alarm clocks instead of one? Explain or show.  
Let W = the alarm works; P(at least one W) = 1 -P(all not W) = 1 - (.01)^3 = 0.999999
The student does not gain much because 0.99 is very close to 0.999999.