Coin Flip

COINFLIP APPLET INSTRUCTIONS
1. Concentrate on the coin landing HEADS up.
2. Click the [FLIP COIN] button to flip your coin.
3. Your coin flip is recorded in the upper left.
4. Repeat steps A to C ten times.
5. When finished, you will have a record of each flip, and
6. The total number of HEADS you successfully willed to occur.
   
7. ScreenCopy your results after 10 flips.
8. Email them to your instructor
9. Along with the answers to the exercises below.

COIN FLIP — PART 1

1. By the fundamental counting principle, the number of ways to toss a coin 10 times is
   
   2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2
   
   How many ways is that?
   
   
2. Is there a shorter way of writing 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 ?
   What is it?
   
3. The number of different arrangements in which you could toss five heads and five tails is the number of possible permutations of the letters in
   
   HHHHHTTTTT
   
   Write an expression with factorials for the number of permutations of these letters.
   Hint: Remember when you counted arrangements of Aces?
   
   Find the value of this expression to find the number of different arrangements in which five heads and five tails can turn up.
   
   
4. The number of different arrangements in which six heads and four tails can turn up is the number of possible permutations of the letters in
   
   HHHHHHTTTT
   
   Write an expression with factorials for the number of permutations of these letters.
   
   Find the value of this expression to find the number of different arrangements in which six heads and four tails can turn up.
   
   
5. Express the number of different arrangements of getting seven heads and three tails in terms of factorials.
   
   Find the number of orders by finding the value of the expression you wrote.
   
   
   
6. There are 10 arrangements of getting nine heads and one tail:
   
   

   HHHHHHHHHT
   HHHHHHHHTH
   HHHHHHHTHH
   HHHHHHTHHH
   HHHHHTHHHH
   HHHHTHHHHH
   HHHTHHHHHH
   HHTHHHHHHH
   HTHHHHHHHH
   THHHHHHHHH
   

   Express the number of arrangements of getting nine heads and one tail in terms of factorials.
   
   Show why this expression is equal to 10.
   
   
7. There is only one arrangement of getting ten heads and zero tails:
   
   HHHHHHHHHH
   
   Write an expression with factorials for the number of arrangements of these letters
   and show that it is equal to 1.
   
   
8. Write down the order of heads and tails that you tossed in the mathlet above.
   How many heads did you get?
   Do you think this qualifies you as telekinetic?
   How many ways can this number of heads come up in 10 tosses?
   Count the ways using factorials.
   
   
9. Now reverse the heads with the tails.
   Everywhere you wrote H, write T; and vice versa.
   How many ways can this number of tails be tossed in 10 tosses?
   Count the ways using factorials.
   
   
   Fill in the table below with your answers from above.
   Use similar thinking to fill in the rest of the table.
   
10. Add the numbers in the Perms column of your table.
    Did you get the number you calculated in exercise 1?
    What is it?

• Screencopy, save as a GIF image, and email your coin tosses to your instructor
• Along with your answers to the questions above.
• Show all your calculations.
• Use the factorial ! symbol in your calculations.
• Screencopy and save as a GIF image, the table above when you have completed it.
• Send your gif images and answers in early and often so your instructor can correct you.
• Re-read these directions. You probably overlooked something.
  Don't lose points unnecessarily.

COIN FLIP — PART 2

x               = Number of heads in 10 tries

10Cx            = Count the ways the coin can land heads x times in 10 tries
                  This is the same as the number of permutations you counted in Part 1.
                  
p^x             = Successive probability of coin landing heads x times in 10 tries
p(Heads)^x

(1-p)^(10-x)    = Successive probability of coin landing tails 10-x times in 10 tries
p(Tails)^(10-x)  

P(x=#Heads)     = Probability of coin landing heads x times in 10 tries

1. Were your guesses dependent or independent of each other?
2. What is the probability of the coin landing heads at most your number of times?
3. What is the probability of the coin landing heads at least your number of times?
4. Are you telekinetic?
   If the probability of the coin landing heads your number of times is less than 10%,
   then the answer is YES;
   otherwise, answer NO!
5. Write a formula for the probability of the coin landing heads P(x) for any n number of coins.
   Use n, x, nCx, and p for symbols.
   P(x) = ?

• Complete the table under EXPECTED RESULTS.
• Screencopy and save the table as a GIF image when you have completed it.
• Answer the questions under OBSERVED RESULTS.
• Show all your calculations.
• Send your gif image and answers in early and often so your instructor can correct you.
• Re-read these directions. You probably overlooked something.
  Don't lose points unnecessarily.
• Part 2 of this web lab is due next Tuesday at noon.