Quantitative MethodsCoin Toss SimulationName: ______
MA 103, Section 01
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, consider the sequence of tosses: H T T T H.
After the first toss, the proportion of heads was 1/1, after the second toss the proportion of heads was 1/2, then after the third toss it was 1/3, 1/4 after the fourth, and finally 2/5 after the fifth toss.
Heads or Tails? / T / H / H / T / T / T / T / H / H / H
Proportion of H so far / 0.00 / 1.00 / 0.50 / 0.33 / 0.25 / 0.40 / 0.50 / 0.57 / 0.63 / 0.56
2. On the following axes, plot the proportion of heads so far, for each toss from your table. What does your graph show?
Initially graph varies with steep rise but as no of flips increases, it starts approaching ½ which is in accordance with law of large numbers.
3. If you wanted to simulate coin tossing in Excel, you could use its =rand() function. rand() produces a random decimal number between 0 and 1. So if we generate a random decimal which is between 0 and 1, and multiply it by 2, the result is between 0 and 2. Then we could take the integer part of this number to get either a 0 or a 1. We can let the 0 represent a “tails,” and 1 represent a “heads.”
So, if rand() generates 0.13061, what value is produced when you multiply it by 2 and take the integer part?
So, if rand() generates 0.7893, what value is produced when you multiply it by 2 and take the integer part?
When rand() generates 0.13061 value is 0
When rand() generates 0.78934 value is 1
4- Write a paragraph explaining what your graph shows
The graph shows that proportion is close to 1/2 as the number of the coins increases which is in accordance with the law of large numbers.
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
The proportion of head changes quite drastically for the first 10 observations when we refresh the graph but the overall probability still remains at .5. There should be enough sample size to check this.
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.)
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."
length of the longest streak of hits – 8
length of the longest streak of misses. – 10
The probability reaches near the value 0.49 in this case which comes close to 0.5 as no of population increases. There is variation in good streak of hitting the shots but when it is averaged out over large population, probability comes close to 0.5.
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
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
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.
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."
length of the longest streak of hits – 9
length of the longest streak of misses. – 3
f. Describe how the "streaks" compare for the 50-percent and 75-percent
As compared to 50% hitting team, 75% hitting team has more probability of hitting the team and less streak of missing.