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In this activity, you will explore some ideas of probability by using Excel to
simulate tossing a coin and throwing a free throw in basketball.
1.
Toss a coin 10 times and after each toss, record in the following table the
result of the toss and the proportion of heads so far. For example, suppose
you obtain the following sequence of heads and tails for the first
five tosses: H T T T H. After the first toss, the proportion of heads so
far is one out of one: 1 or 1. After the second toss, the proportion of
1heads so far is one out of two: 1. After the third toss, the proportion
2of heads is one out of three: 1. After the fourth toss, the proportion of
3heads is one out of four: 1. After the fifth toss, the proportion of heads is
4
two out of five: 2
2
.5
Toss # 1
H or T?
Prop of H So Far
3
4
5
6
7
8
9
10
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2.
On the following axes, plot the proportion of heads so far, for each toss from
your table. What does the graph show?
Proportion
20468101.00.80.60.40.20Proportion of Heads
Toss
Now, you will use Excel to simulate 1000 independent tosses of a fair coin and
plot on a graph the proportion of heads so far after each toss using the instructions
that follow #3.
3.
In Excel, the function RAND() (that is, RAND followed by two parentheses)
produces a decimal number between 0 and 1, in such a way that every decimal
number between 0 and 1 is equally likely to be produced. You will use the
RAND() function to generate integers 0 or 1 with equal probability. The integer
1 will signify “heads” and the integer 0 will signify “tails.” To get a 0 or 1
with equal probability, you’ll multiply the random number by 2 and then take
the integer part of it; that is, you will drop all digits after the decimal point.
Suppose the decimal number produced is 0.13061. What value do you
get if you multiply that number by 2 and then take the integer part of it?
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Suppose the decimal number produced is 0.78934. What value do you
get if you multiply that number by 2 and then take the integer part of it?
The Excel formula to do this is =INT(2*RAND()).
Instructions to Use Excel to Simulate
Tossing a Coin
a.
Open Excel and start with a blank worksheet. Enter the label Results from
1000 Tosses in cell A1.
b.
Enter the formula =INT(2*RAND()) in cell A3. Drag this formula down to
cell A1002 to generate a column of 1000 0s and 1s, representing 1000 tails
and heads.
c.
Enter the label Heads, So Far in cell B1. In cell B2, enter the value 0, and in
cell B3, enter the formula =A3+B2. (This formula will keep a running count
of the number of heads so far.) Drag this formula down to cell B1002.
d.
Because you want to keep a running count of the proportion of heads, you’ll
start by recording the number of tosses so far. In cell C1, enter the label
Tosses, So Far. In cell C3, enter 1, and in cell C4, enter 2. Highlight cells
C3 and C4; then drag down to cell C1002. You should have a column of
integers 1 through 1000.
e.
Now fill the cells D3 to D1002 with the value 0.5, so the graph you construct
will have a horizontal line at the height 0.5. (Do this efficiently using autofill.)
f.
In cell E1, enter the title Proportion of Heads, So Far, and in cell E3 enter
=B3/C3. Drag this formula down to cell E1002.
g.
To display the results of the coin toss simulation on a graph, click the
Chart wizard button. Select Line for Chart type and Line for Chart
sub-type (this will be the first sub-type choice). Click Next.
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h.
Now (in step 2 on your Excel screen) for the Data range enter D3:E1002, or
highlight these cells and select the Series in: Columns button. Click Next.
i.
For this next step, on the Titles tab, enter a graph title and labels for the
axes. On the Axes tab, select Automatic for Category (X) axis and place
a check mark in the Value (Y) axis box. On the Gridlines tab, turn off all
gridlines, and on the Legend tab, clear the legend.
j.
Click Finish to complete the graph and format the X and Y axes as you
choose.
4. Write a paragraph explaining what your graph shows.
5.
Put the cursor in any blank cell near your graph. Press Ctrl= to change the
random numbers and your graph. Do this several times and describe how the
graph changes.
6.
Shaquille O’Neal is not particularly good at free throws. He makes about
50 percent of his free throws over an entire season.
a.
Go to sheet 2 and set up a new worksheet to simulate 100 free throws shot
independently by a player who has probability 0.5 of making each shot. In
column A, keep a record of the attempt number by generating integers 1,
2, . . . , 100 in cells A2 through A101. (Enter an appropriate title in cell A1.)
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s
b.
Label cell B1 Hit or Miss and generate a random sequence of 1s (each 1
represents a hit) and 0s (each 0 represents a miss) in cells B2 through B101.
c.
Compute the overall proportion of hits by calculating the mean of the
B column values. Also look at your data and identify the length of the
longest streak of hits and the length of the longest streak of misses.
Write a paragraph commenting on the proportion of hits and the “streaks.”
7.
A certain college’s men’s basketball team is quite accomplished at making free
throws. According to the coach, in their most recent season the team made
approximately 75 percent of free-throw attempts, and ranked 10th in the country
among similar schools in successful free-throw attempts.
a.
Set up another new worksheet to simulate 100 free throws shot by team
members who have probability 0.75 of making each shot.
b.
Again, use column A to keep track of the attempt number in cells A2
through A101.
c.
Label cell B1 Hit or Miss. Now you will generate a sequence of equally
likely occurrences of the numbers 0, 1, 2, or 3 with 0 representing a
miss and any of the other three numbers representing a hit. By entering
the formula =INT(4*RAND()) into cell B2, you can set the sequence up
so each of the numbers occurs with equal probability. Then autofill to
cell B101.
d.
In cell C2, enter the formula =IF(B2=0,0,1). This will give you a “1” in cell
C2 if cell B2 recorded a hit, and a “0” if cell B2 recorded a miss. Autofill
the formula down to cell C101.
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e.
Find the overall proportion of hits, and identify the length of the longest
streak of hits and the length of the longest streak of misses. Write a paragraph
commenting on your proportion of hits and your “streaks.”
f. Describe how the “streaks” compare for the 50-percent and 75-percent
scenarios.
Summary
In this activity, you learned how to use Excel to simulate random processes.
You simulated the random process of tossing a coin and graphed the frequency
of heads to visualize the probability that the toss comes up heads. You also used
Excel to simulate basketball free throws and analyzed the proportion of hits and
streaks of hits and misses.
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