Probability of Tossing a Coin and Law of Large Numbers

Probability of Tossing a Coin and Law of Large Numbers

Executive Summary

To demonstrate the evaluation and calculation of the classical probability of tossing a coin three times and the application of the Law of Large Numbers, this project was carried out. In this case, an unbiased coin i.e., P(Getting a head)= P(Getting a tail)= 0.5, is tossed n times, can we exactly acquire that n/2 heads and n/2 tails? In case the number of n is small, then it will be impossible to acquire that. Though a fair coin will not be able to exhibit 50% of heads  and 50% of tails respectively. However, if the number of trials n is big enough, then according to the law of large numbers, it can be deduced that the empirical probability of acquiring either a tail or a head could be closer to 0.5. However, on tossing a coin 3 times there is a probability that we could acquire either three heads, two heads and one tail, at least one tail or at least two head from the activity. The results of the toss can be generated from at least 500 outcomes through excelle and calculating the results of the empirical probabilities of this project.

Introduction

When we toss a coin there are only two possible outcomes, either a head or a tail.  It can be either a head or a tail which have equal likelihood of occurrence. Probabilities are always indicated in zero and ones. A probability represented by one indicates that the occurrence of the event is quite certain. In case a coin is tossed, then it will either come up as a head or a tail. There is always a probability that either one of them can happen. A probability of zero indicates that the occurrence of event is quite impossible. When a coin is tossed, it would be impossible to acquire both head and the tail at the same time. Before carrying out this probability test, it is important to take into account a number of essentials. Most importantly, probabilities do not account or tell us about what is going to happen, they merely indicate what is going to happen. It is quite unlikely that if we toss a coin thrice, that all the results will come out as heads.

Working Out Probabilities by Counting

On tossing a coin thrice, the following are the possible outcomes; TTT, TTH, THT, THH, HTT, HTH, HHT, HHH. All these outcomes vary from each other and all of them have equal likelihood of happening. There are only three tosses with only one head, TTH,THT and HTT. Therefore, the probability is 3/8 and converting this fraction into decimal we acquire 0.375 or a 37.5% percentage. In order to acquire the correct probability of the toss, we acquire the desired outcomes divided by the total number of outcomes.

Tossing Coins Theory

As discussed in the section above, probabilities of a coin are represented in zeros and one. If a coin is tossed, it will either come out as head or tail and there is a probability that either one of them could happen. A probability of zero indicates that there is zero probability of an event to happen. According to the Tossing Coins Theory,   P(H)= P(T)= 0.5. Therefore, in order to acquire the probability of an event to happen, the following theoretical formula is used;

P((H/T)/ A)= P(AnB)/ P(A) where A refers to the total outcomes acquired from the toss.

Discussion

In order to understand the concept of probability, it is important that we use an example to understand. In case one unbiased coin is tossed, what is the probability of getting all heads as outcomes, getting two heads, getting at least one head, getting two heads and getting at most two heads?

The above question can be solved with the help of a probability tree diagram and probability space.

In tossing a coin three times, there are 8 possible outcomes. Some of these outcomes include;   {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

Using a probability tree we can summarize the outcomes in three simple steps.

Step 1

On tossing the coin the first time, we acquire two outcomes, a head (H) and a Tail (T)

Step 2

On tossing the coin the second times, four outcomes are acquired. In this case, the probability outcome is (2)²=4 With the help of a probability tree, and a sample space table we will illustrate the outcomes of tossing a coin the second time.

Probability tree diagram

Sample Space table

The sample space table and the probability tree diagram above indicate the possible outcomes that can be acquired from tossing a coin the second time.

Step 3

On tossing the coin the third time, eight outcomes are acquired. In this case, the probability of the outcome is (2)³= 8 With the help of a probability tree and a sample space table, we will illustrate the outcome of the tossing the coin the third time.

Probability Tree diagram

Sample Space Table

The above three steps, illustrate a systematic approach of acquiring the eight outcomes of tossing a coin three times. Using the above diagrams , let’s now answer the questions.

• Getting all heads from tossing the coin three times

Let E1= Event of acquiring all heads

Then, E1= (HHH) and n(E1) = 1

Applying the probability formula, the probability of getting all heads as the outcome is

P(getting all heads)= P(E1)= n(E1)/n(S)

=1/8

Therefore the probability of acquiring three heads from tossing a coin three time is 1/8, in decimal form the probability is 0.125

• Getting two heads from tossing a coin three times

Let E2= Event of getting two heads.

Then, E2= (HHT, HTH, THH)

And therefore, n(E2)=3

Therefore, P(getting 2 heads) = P (E2)= n(E2)/n(S)=3/8

The probability of acquiring two heads from tossing a coin thrice is 3/8, in decimal form, the probability is 0.375

• Getting one head from tossing a coin three times

From the outcomes acquired from the three steps evaluated above, it is possible to evaluate the probability of getting one head from the outcomes.

Let E3 be an event of acquiring one head.

Then, E3= (HTT, THT, TTH) and therefore, n(E3)= 3

Applying the probability formula we evaluate that, P(getting one head)= P(E3) = n(E3)/n(S)= 3/8

The probability of acquiring one head from tossing a coin thrice is 3/8, in decimal form, the probability is 0.375

• Getting at least one head

The probability of acquiring at least one head can be evaluated from the above outcomes;

Let E4= (HTT, THT, TTH, HHT, HTH, THH, HHH) and therefore, n(E4)= 7

Therefore, by applying the probability formula, we acquire that P(getting one head) = P(E4) = n(E4)/n(S)= 7/8, in decimal form the probability is 0.875

• Getting at least two heads

If we let E5 be the event of acquiring at least 2heads, then E5=

(HHT, HTH,THH, and HHH) and therefore, n(E5)=4

Applying the probability formula, we find out that the P(getting at least 2 heads)= P(E5)= n(E5)/ n(S)=4/8=1/2

The probability of getting at least two heads from tossing a coin three times is ½, in decimal form its 0.5

• Getting at most two heads

If E6= an event of getting at most two heads, then, E6= (HHT, HTH, HTT, THH, THT, and TTT) and therefore, n(E6) = 7

Applying the probability formula, the probability of getting at most two heads= P(E6)= n(E6)/n(S)=7/8

The probability of getting at most two heads is 7/8 and in decimal form, the probability is 0.875

Pascal Triangle of Probability

A probability tree diagram and sample space table as described above are some of the effective ways of evaluating the probability of tossing a coin. However, Blaise Pascal introduced the Pascal’s Triangle principle to evaluate on the probability of events. The Pascal’s triangle works using the principle of addition. Each and every line within the triangle is composed of adding two numbers in the line above. In this case, you assume that there are zeros and the beginning and at the end of each line and you start with the one on the top row. According to Pascal, the Pascal’s triangle is an approach that crops out a lot of probability. For instance, if we take the nth row of the Pascal’s triangle, and the Mth number in it, and you have the top bit (numerator) of the probability of determining the m heads when tossing n coins. Below is a diagrammatical representation of the Pascal’s triangle.

Conclusion

Tossing a coin and determining its probability is classified as systematic random sampling. This form of sampling is an improvement of simple random sampling. While determining the probability of a coin, there are some of the key things that one should consider. Firstly, it is important to note that the probabilities are not performed to know what is going to happen, but rather they merely indicate what is likely to happen. Before carrying out this probability test, it is important to take into account a number of essentials. Most importantly, probabilities do not account or tell us about what is going to happen, they merely indicate what is going to happen. It is quite unlikely that if we toss a coin thrice, that all the results will come out as heads.

It is quite unlikely that if one tosses a coin finite number of times, that the number of heads and the number of tails turn out to be the same.

References

Andale. (June 26, 2015). Probability Sampling: Definition,Types, Advantages

and  Disadvantages.  Statistics  How  To.   Retrieved

from http://www.statisticshowto.com/probability-sampling/.

Babbie, E. R. (n.d.). The Logic of Sampling. The Basics of Social Research.

Wadsworth Cengage Learning, pp 208.

Chaudhuri, A., &Stenger, H. (2005). Survey Sampling: Theory and Methods –

2nd ed. Chapman & Hall/CRC.

Daniel,  J.  (2012).  Sampling  Essentials:  Practical  Guidelines  for  Making

Sampling Choices. Sage Publications, pp 103.

Doherty,  M. (1994) Probability versus Non-Probability Sampling in Sample

Surveys, The New Zealand Statistics Review March 1994 issue, pp 21-28.

Fink,  A. (2003) How  to Sample in Surveys.  2nd Edition. Thousand Oaks:

Sage.

Henry, G.T. (n.d.). Practical Sampling. Sage Publications, pp 23.

King,  R.  M.  Types  of  sampling.  Advanced  Research  Methods.  Retreived

from http://www.psyking.net/HTMLobj-3829/Types_of_Sampling.pdf.

Lynchi,  G.  (n.d.).  Sampling.  Retreived

from https://www.kent.ac.uk/religionmethods/documents/Sampling.pdf.