# Coin Toss Probability

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. 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: _ 11_ or 1. After the second toss, the proportion of heads so far is one out of two: _12_ . After the third toss, the proportion of heads is one out of three: _ 13_ . After the fourth toss, the proportion of heads is one out of four: _14_ . After the fifth toss, the proportion of heads is two out of five: _ 25_ .

Toss # 1 2 3 4 5 6 7 8 9 10

H or T?

Prop of H So Far

1.

SSAc18.

https://brainmass.com/statistics/probability/251620

#### Solution Preview

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. 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: 1/1 or 1. After the second toss, the proportion of heads so far is one out of two: 1/2. After the third toss, the proportion of heads is one out of three: 1/3. After the fourth toss, the proportion of heads is one out of four: 1/4. After the fifth toss, the proportion of heads is two out of five: 2/5.

Toss # 1 2 3 4 5 6 7 8 9 10

H or T? T H H T T T T H H H

Prop of H so far 0.00 0.50 0.67 0.50 0.40 0.33 0.29 0.38 0.44 0.50

The above graph shows us that the proportion of heads at the beginning of the trials is more variable than at the end. As the number of flips increases, it seems that that proportion of heads so far approaches ½. This seems to be born out in the above chart.

Activity 18.1, question 3, p. 577-578 (Sevilla, A., & Somers, K., 2007).

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 in the table "Instructions to Use Excel to Simulate Tossing a Coin."

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?

You would get a 0.

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 ...

#### Solution Summary

The solution examines the coin toss probability.

Classical and Empirical Probability - Coin Tossing

1. Describe two main differences between classical and empirical probabilities.

Gather coins you find around your home or in your pocket or purse. You will need an even number of coins (any denomination) between 16 and 30. You do not need more than that. Put all of the coins in a small bag or container big enough to allow the coins to be shaken around. Shake the bag well and empty the coins onto a table. Tally up how many heads and tails are showing. Do ten repetitions of this experiment, and record your findings every time.

State how many coins you have and present your data in a table or chart.

Consider just your first count of the tossed coins. What is the observed probability of tossing a head? Of tossing a tail?

Show the formula you used and reduce the answer to lowest terms.

Did any of your ten repetitions come out to have exactly the same number of heads and tails? How many times did this happen?

2. How come the answers to the step above are not exactly 0.5 and 0.5?

3. What kind of probability are you using in this "bag of coins" experiment?

Compute the average number of heads from the ten trials (add up the number of heads and divide it by 10).

4. Change this to the average probability of tossing heads by putting the average number of heads in a fraction over the number of coins you used in your tosses.

5. Did anything surprising or unexpected happen in your results for this experiment?

Write the sample space for the outcomes of tossing three coins using H for heads and T for tails.

What is the probability for each of the outcomes?

Which kind of probability are we using here?

6. How come we do not need to have three actual coins to compute the probabilities for these outcomes?

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