4332.sagews

#    First, create a "first head" random variable, which simulates flipping a coin until it is heads,
#    and records which flip is the first head.


def first_head():
    coin=0    # 0 for tails, 1 for heads.  Start by setting the coin to tails
    count=0   # Keep track of the number of flips before first head 
    while coin == 0:    # the two lines in this "while loop" will be executed as long as the coin value is 0
                        # but once it equals 1, the loop ends and we move to the final line of the function def.
        coin=randint(0,1)    # set coin equal to 0 or 1 randomly
        count += 1           # increment the count
    return count             # this is the count value when the coin first equals 1 (and thus, the while loop ends).
    
# test a few outputs and compute the average payoff (which, theoretically, is INFINITE).
sum=0
nTrials=10
for _ in range(nTrials):
    fh = first_head()
    print('First head at flip # %s.  You win %s dollars!'%(fh, 2^fh))
    sum += 2^fh
print('Average payoff was %s.' %float(sum/nTrials))

First head at flip # 1. You win 2 dollars! First head at flip # 2. You win 4 dollars! First head at flip # 3. You win 8 dollars! First head at flip # 5. You win 32 dollars! First head at flip # 1. You win 2 dollars! First head at flip # 3. You win 8 dollars! First head at flip # 1. You win 2 dollars! First head at flip # 1. You win 2 dollars! First head at flip # 5. You win 32 dollars! First head at flip # 3. You win 8 dollars! Average payoff was 10.0.

################################################################################
#    Here is one simulation of playing 1000 games, where the price to play each
#    game is 5 dollars.  The result is a plot of the money on hand after each play,
#    using the list_plot function.
################################################################################    

price = 5    #cost to play the game once
nGames = 1000
accumulation = 0    #player's profit
sequence = []       #keep track of the first_head() outputs
daily = []          #accumlation on each day; to be built
for _ in range(nGames):
    fh = first_head()
    sequence.append(fh)
    accumulation += 2^fh-price
    daily.append(accumulation)
    
list_plot(daily,plotjoined = True)

#########################################################################################
#    This is an INTERACTIVE simulation which allows you to change the price per game
#    and the total number of games.  It also prints out the maximum payout, and 
#    when this happened, the final amount of money on hand, the minimum amount, and the maximum,
#    and the average payout (which, theoretically, is INFINITY). 
#    
#    Note the use of the @interactive command, followed by a function definition (the
#    name of the function can be anything, but we choose the generic "_" since we will never use the 
#    function's name.  You can look up the use of Sage interactive online, or just study this example.
#    It's pretty flexible, and easier to use than Mathematica's Manipulate command, although less slick.
##############################################################################################

# We include the already-defined first_head function, so that this is a stand-alone cell.

def first_head():
    coin=0    # 0 for tails, 1 for heads.  Start by setting the coin to tails
    count=0   # Keep track of the number of flips before first head 
    while coin == 0:    # the two lines in this "while loop" will be executed as long as the coin value is 0
                        # but once it equals 1, the loop ends and we move to the final line of the function def.
        coin = randint(0,1)    # set coin equal to 0 or 1 randomly
        count += 1           # increment the count
    return count             # this is the count value when the coin first equals 1 (and thus, the while loop ends).
 
 
 
    
############################################################################################################
#    interactive part starts here
############################################################################################################

@interact
def _(price= [5,10,15,20], nGames = [100,1000,10000,100000]):
    # note that the selection boxes for price and nGames are defined in the function defn above.
    accumulation = 0    #player's profit
    sequence = []       #keep track of the first_head() outputs
    daily = []          #accumlation on each day; to be built
    for _ in range(nGames):
        fh = first_head()
        sequence.append(fh)
        accumulation += 2^fh-price
        daily.append(accumulation)
      
# Compute average payout  
    totalPayout = 0
    for i in range(len(sequence)):
        totalPayout += 2^sequence[i]
    avg = float(totalPayout/nGames)
        
# Compute longest run of tails, max min, etc.    
    longestRun=max(sequence)
    longestRunIndex=sequence.index(longestRun)
    print('Luckiest: game # %s, first head at position %s, winning $%s.'%(longestRunIndex, longestRun, 2^longestRun))
    print('Lowest: %s'%min(daily))
    print('Highest: %s'%max(daily))
    print('Final: %s'%daily[nGames-1])
    print('Average payout:  %s.'%avg)

# Plot the money on hand.    
    l=list_plot(daily,plotjoined = True,gridlines=True)
    show(l)    #in interactive mode, the plot doesn't display with just the list_plot() command.
               #instead, we use the show() function.

price  nGames  [removed] [removed] [removed] [removed]