Calculate the theoretical discrete probability
distribution of number of heads in 20 tosses of a coin.
Theoretical probability distribution gives an idea of an ideal
probability distribution, what a distribution should be given the parameters. Tossing
of a coin is an example of the two-category discrete probability distribution
of sampling with replacement. Refer to my earlier Statistical Notes for clarity
on calculating the two-category discrete probability using tree diagram,
formula and Excel software function.
Why 20 tosses were chosen in this example?
One may pose why 20 tosses were chosen, why not other numbers. Tossing
of an unbiased coin for ‘n’ independent trials is a case of Binomial
Distribution, which is symmetric when the sample size is even number. When ‘n’ is large, Binomial distribution tends to
Normal Distribution with the mean number of successes equal to ‘np’. A guide is
that the mean should be at least five. For an unbiased coin, the number of
independent trails should be at least 10 to get the mean value of five. 20
tosses are more than minimum number of trials required for the approximation to
the Normal Distribution. Second, why I did not choose other numbers than 20
tosses? If I choose 20 tosses, the mean number of success will be 10, a number
between not even a single head and all 20 heads turn up, which symmetrically or
equally divides the numbers of head to the lesser and greater sides.
Theoretical Discrete
Probability Distribution
The probability of
success that a head lands in a toss of a coin denoted by ‘p’ is equal to half. Unlike,
the probability of failure, the tail turns up, denoted by ‘q’ is also half.
Let X be a random variable of interest, successful event
that the head lands up in a toss of a coin. X takes one of the values
from not even a single head to all 20 heads in 20 tosses, denoted by ‘x’.
Because, among 20 tosses of a
coin it is likely that the number of heads that turns up could vary between not
even a single head to all 20 heads. If the tail turns up in every toss of a coin,
then the total number of heads is zero. That will happen if the coin is fully
biased to the tail. Unlike, if the head turns up in every toss, the total
number of heads is 20, which is again fully biased to the turning up of the
head. These two are most extreme cases. In other instances, the number of heads
that turn up will vary between these two extreme numbers.
Binomial distribution function is used to calculate the theoretical two
category discrete probability distribution of this examples. The probability distribution of X
depends on the index ‘n’ and parameter ‘p’, and is given by the expression: P(X=x)
= C(n,x)pxqn-x. One can calculate the theoretical
probability distribution in different ways.
Way one: One way is to count the number of possible outcomes for a category of
outcomes and the total number of outcomes. The
total number of outcomes is calculated using the formula ‘number of possible
outcomes of a trial power the number of trials’. In this example, the total
number of outcomes is calculated to be, ‘2n’, which is 220,
equal to 1,048,576.
The number of outcomes for a certain outcome type can be calculated
using the Binomial coefficient value. If I am interested to know the number of
outcomes constituting 10 head and 10 tails in 20 tosses, I can calculate it
using the Binomial coefficient C(20,10), equal to 184756.
The probability of 10 heads in 20 tosses is calculated by number of
outcomes for 10 heads in 20 tosses divided by total number of outcomes, 184756
divided by 1,048,576, equal to 0.176197. Likewise, the number of possible
outcomes for each outcome category is calculated as presented and highlighted
yellow in Table 1.
Way Two: Using the Binomial formula, example, for 10 heads in 20 tosses is
calculated as C(20,10)(0.5)10(0.5)10, equal to 0.176197, highlighted yellow in Table 1. This formula can be applied
to all possible number of heads in 20 tosses to manually calculate the
probability distribution.
Third Way: Excel software
function can be used for this example. The total probability of not even a
single head to all 20 heads turned up in 20 tosses of a coin is one. Turning up
of 10 heads out of 20 tosses has the highest probability, denoted by P(X=10)
equal to 0.176197 highlighted yellow in Table 1.
Table 1: Type, Number and
Probability of Outcomes 20 tosses of a coin
Graphical
Presentation
Chart 1 shows the theoretical two category discrete
probability distribution of number of heads in 20 tosses of a coin. This clearly shows the bell-shaped curve, the
symmetric line chart of theoretical probability distribution.
Conclusion
Looking at the table and the chart one can see that turning up of 10 head in 20 tosses is highly likely. This is
because as I discussed above the mean value is 10. The likelihood decreases
towards both sides of the mean value. Two extreme number of heads, 0 and 20,
have the least chance of occurrence. Since the number of toss is sufficiently
more than the minimum number of 5 tosses, the Binomial distribution is approximate
to the Normal distribution.
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