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

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

You are an artillery officer. Your gun crew has a 90% probability of directly hitting the target when it fires a shell. How many shells do you have to fire at a target in order to have a 99% chance of making a direct hit?

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

Let p be the probability that the shell is a direct hit.

a.What is the value of in the binomial calculator?

b.What is the value of q?

c.What is the value of n that should be used to have a 99% probability of getting a direct hit? (include a screen shot of your answer)

d.What is the probability of getting a direct hit when firing 3 shells? (include a screen shot of your answer). Remember to hit the "Recalculate" button and to check to see if you picked "Above", "Below", or "Between".

2.(5 points) You are the statistician for an airline which still engages in the practice of "overbooking". There are 300 seats on a particular flight. From past data, you know that on average, only 94% of people who pay in advance for a ticket actually show up. (They are sick, get caught in traffic, are delayed by weather, sudden change of plans, or just by a ticket because they don't know their schedule, etc.)

What is the number of tickets (N) that you should sell so that you have only between a 1% and 2 % chance of going over the 300 seat limit?

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

Let the probability of a person buying a ticket actually showing up be p, and q be the probability of the person not showing up.

a.What is the value of p?

b.What is the value of q?

c.Using the binomial calculator, select the button "above" and put in 300.

d.The value of N is the number of tickets you sell. What is the probability that more than ("above") 300 people will show up if you sell only N = 300 room reservations? (Make a "reasonable" guess.)

e.Using the binomial calculator, do you get the same answer that you guessed in (d)? (yes or no)

f.Using the binomial calculator, gradually increase the value of N until the probability of getting more than 300 people showing up is between 1% and 2% (include a screen shot of your answer). What is this number N?

3.(5 points) Calculate the odds of winning the Power Ball $50,000 prize. In order to win, you must match only 4 of the 5 winning white balls numbered 1 to 69, and match the red power ball numbered 1 to 26. Write your final answer as a ratio instead of decimal fraction. For example, write instead of 0.25. 1/10 instead of 0.10. Note: There are 2 separate bags - one with the 69 white balls and another with the 26 red balls. For the white balls, we are picking without replacement. (Note: If necessary, draw a bag with 69 balls in it and mark 5 balls as the winning balls. As you draw each ball out of the bag, cross out that ball in your bag. Draw another bag with 26 red balls and mark one ball as the winning ball.)

a.What is the probability of picking any of the 5 winning white balls from the bag for your 1^stpick?

b.What is the probability of picking any of the remaining 4 winning white balls from the bag for your 2^ndpick?

c.What is the probability of picking any of the remaining 3 winning white balls from the bag for your 3^rdpick?

d.What is the probability of picking any of the remaining 2 winning white balls from the bag for your 4^thpick?

e.What is the probability of picking a losing (NOT picking the 1 remaining winning) white ball from the bag for your 5^thpick?

f.Now we could have also picked the losing ball 1^st, 2^nd, 3^rd, or 4^th, not just 5^th. Therefore, we have to multiply our probability by 5C1 since order does not matter in picking the 4 winning white balls and the 1 losing white ball. What extra number do we have to include in our probability calculation?

g.What is the probability of picking the winning red power ball from the other bag?

h.Using the multiplication rule for the numbers you got for a-g, what is the probability of winning the Power Ball lowest prize? (Note: The Power Ball ticket says the probability of winning are 1/913,129. If you do not get something close to this number, then you did something wrong!)

4.(10 points) In a basket ball game, players attempt 2 point shots 66.67% of the time and 3 point shots 33.33% of the time. If a 2 point shot is attempted, 75% of the time there is a score while 25% of the time there is a miss. If a 3 point shot is attempted, 55% of the time there is a score while 45% of the time there is a miss.

Fill in the values to the following tree diagram. (use 4 digits).

.AAAA =.CCCC =.GGGG =

.DDDD =.HHHH =

.BBBB =.EEEE =.IIII=

.FFFF =.JJJJ=

a.What is the probability of attempted shots which result in a 2 point score?

P(2 pt score) =

b.What is the probability of attempted shots which result in any score (2 point or 3 point)?

P(score) =

c.If an attempted shot is a score, what is the probability that it was a 2 point attempt?

P(2 pt attempt|score) =

d.If a 2 pt shot is attempted, what is the probability that it results in a score?

P(score| 2 pt attempt) =

5.(10 points) Determine the poker odds of drawing the following hand from a standard card deck. (4 suits, 13 ranks in each suit.)

What are the odds of drawing 1 pair (2 cards of one rank and 3 other cards which do not match in rank the pair or each other, or WWXYZ)? Drawing cards is without replacement. (NOTE: If necessary, lay out 52 cards on a table and do a dry run before computing the probabilities!)

In order to get a full house, you must pick the following 5 cards:

a.What is the probability that you will pick a W for your first card?

b.What is the probability that you will pick a matching W your second card?

c.What is the probability that you will pick a non matching X for your third card?

d.What is the probability that you will pick a non matching Y for your fourth card?

e.What is the probability that you will pick a non matching Z for your fifth card?

f.Now we picked the cards in the order

WW X Y Z

We could have also picked the cards in many other orders such as

W W Y X Z,W Z W X Y, etc.

Therefore you have to multiply your answer by a factor F to get the correct probability. This factor is

F =, where 5 is for the 5 cards, 2 is for the pair, and the 1's are for the 3 non-matching cards.

g.Multiply the above numbers (a) to (f) to get the final probability.

(Note: The probability of getting a pair is 1/2.37. If you did not get close to this number, then you did something wrong.)

6.(10 points) Use the following values to fill in the Venn diagram.

Circle A = "2" + "3"

Circle B = "3" + "4"

football C = "3" = overlap of A and B

Total = area("1") + area("2") + area("3") + area("4")

NOTE: "1", "2", "3", and "4" doNOToverlap each other, but circles A and B do overlap each other.

Let A be the number of students taking math = 900.

Let B be the number of students taking classics = 600.

Let C be the number of students taking both math and classics = 200.

Let the total number of students in the school be 1500.

a.What is the value of area("3") the overlap?

b.What is the value of area("2") the part of circle A which does not overlap circle B?

c.What is the value of area("4") the part of circle B which does not overlap circle A?

d.What is the value of area("1") the part outside of the 2 circles?

e.What is the value of P(A) = ("2" + "3")?

f.What is the value of P(B) = ("3" + "4")?

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

h.What is the value of P(A U B)?

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

j.Are the events A and B "mathematically independent" in the mathematical sense? (yes or no, and explain your answer) Hint: Compute P(A)*P(B), and P(AB) and compare the two numbers you get.

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

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

7.(10 points) Fill in the following table when using 2 dice.

1^stroll / 2^ndroll

1

2

3

4

5

6

1

2

3

4

5

6

a.Let event A be that the 1^stdie roll is a 1 or 6. Put an A in the appropriate boxes. It does not matter what the 2^nddie roll is for event A.

b.Let event B be that sum of the 2 die rolls = 6 (5+1, 4 +2, etc). Put a B in the appropriate boxes.

c.What is P(A)?

d.What is P(B)?

e.What is P(A B)?

f.What is P(B|A)?

g.What is P(A|B)?

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

i.Are events A and B mathematically independent? (yes or no) Be sure to give your reasoning. Hint: Compute P(A)*P(B), and P(AB) and compare.

8.(10 points) Fill in the following contingency table to answer this question.

Boys (B)

Girls (G)

Total

Sports fan (S)

300

Music lover (M)

800

Total

500

1000

a.If a person is chosen at random, what is the probability that they will be a boy, P(B)?

b.If a girl is chosen at random, what is the probability that she will be a music lover, P(M|G)?

c.If a sports fan is chosen at random, what is the probability that they will be a boy, P(B|S)?

d.If a person is chosen at random, what is the probability that they will be a music lover boy OR a sports fan girl, P[(M B) U (S G)]?

9.(10 points) You are playing a gambling game with a 12-sided die. If you roll an odd number, then you lose $6. If you roll an even number, then you win that amount in dollars (i.e., you roll a 2, you win $2, etc). What is the Expected average winnings/losings of this game?

x = die roll

P(x)

Payoff(x)

P(x)*Payoff(x)

1

2

3

4

5

6

7

8

9

10

11

12

E =