(Binomial distribution / Pascal’s triangle / Binomial probability)
In this section, the three interesting concepts discussed are binomial distribution, Pascal’s triangle, and binomial probability.
1. Binomial distribution:
“We can get a better feeling for the details of these results if we plot a graph of the distribution of the results (Feynman et al., 1963, section 6.2 Fluctuations).”
Dr. Sands initially asks how many “heads” we are expected to get if a coin is tossed N times. To illustrate the concept of binomial distribution, 100 experiments were actually done by shaking 30 coins violently in a box and then the number of heads was counted. As a result, there were 3000 tosses of coin during the experiments and the number of heads obtained was 1493. Simply put, the fraction of tosses that gave heads is 0.498 which is slightly less than half. Thus, we do not assume that the probability of throwing heads is greater than 0.5. However, it is possible to have a fluctuation from the binomial distribution such that one particular set of observations of 30 coins gave 16 heads most often instead of 15 heads.
A binomial distribution can be related to a repeatable experiment or observation that has two different outcomes. This distribution is important in physics because it can be used to describe a Bernoulli process, such as a random walk of a molecule in one dimension. There are four important properties of a binomial distribution: (1) There is a fixed number of repeatable trials, for example, toss a coin 30 times. (2) The trials are independent of each other, i.e., the result of one trial has no influence on any other trial. (3) The probability of an outcome denoted by p remains constant throughout the experiment. (4) There are two possible outcomes, such as a “head” and a “tail.” (Remember “fict.”)
2. Pascal’s triangle:
“…The set of numbers which appears in such a diagram is known as Pascal’s triangle. The numbers are also known as the binomial coefficients because they also appear in the expansion of (a + b)n (Feynman et al., 1963, section 6.2 Fluctuations).”
Dr. Sands explains that the number of “ways” of getting different numbers of heads and tails can be illustrated by the set of numbers in Pascal’s triangle. The numbers are also known as the binomial coefficients (or combinatorial numbers) because they are the coefficients of the xk term in the polynomial expansion of (a + b)n. If n is the number of tosses and k is the number of heads obtained, then it can be represented as C(n, k), nCk, or nCk. In general, the binomial coefficients can be computed from C(n, k) = n!/k!(n−k)!, in which n! is commonly called “n-factorial” that represents the product (n)(n−1)(n−2)…(3)(2)(1). In essence, the expression C(n, k) is about different combinations of k “heads” that could occur in a sequence of n tosses without considering the order of permutation (or redundancies).
Note: There are claims that the triangle of binomial coefficients was discovered earlier by mathematicians in India, Greece, Iran, China, Germany, and Italy. For example, Yang Hui (1238 – 1298) presented the binomial coefficients in his book, titled Xiangjie Jiuzhang Suanfa (Needham, 1959, p. 135). More importantly, he acknowledged that his method of finding square roots and cubic roots by using these numbers was discovered earlier by another mathematician Jia Xian (1010 – 1070). Jia Xian’s book entitled Shi Suo Suan Shu was written about 600 years before Pascal (1623 – 1662).
3. Binomial probability:
“…This probability function is called the Bernoulli or, also, the binomial probability (Feynman et al., 1963, section 6.2 Fluctuations).”
Dr. Sands elaborates that we can compute the probability P(k, n) of throwing k heads in n tosses by using the definition of probability mentioned earlier. Firstly, the total number of possible outcome in n tosses is 2n because there are 2 outcomes for each toss. Next, the number of ways of obtaining k heads from n tosses is C(n, k) and thus, the probability P(k, n) is equal to C(n, k)/2n. In general, we can designate the two outcomes by W (for “win”) and L (for “lose”) in which the probability of W or L in a single trial may not be equal to 0.5. If we let p be the probability of obtaining the result W, then the probability of L is (1−p) and it can be represented as q. In short, the probability P(k, n) that W will be obtained k times in n trials is P(k, n) = C(n, k) pkqn−k.
Curiously, Dr. Sands simply calls this probability function the Bernoulli or the binomial probability. However, the Bernoulli distribution is commonly known as a special case of the binomial distribution in which there is only one trial (n = 1). In other words, we can use the binomial distribution to find the probability of recurring Bernoulli trials. Importantly, if every Bernoulli trial is independent, then the number of “win” in Bernoulli trials has a binomial distribution. Furthermore, some may use 1 and 0 to represent “head” and “tail” (or vice versa) in a coin toss.
Questions for discussion:
1. What are the properties of a binomial distribution?
2. How are binomial coefficients related to the number of “ways” of getting certain numbers (k) of heads in a number (n) of trials without considering the order of heads (k!) and tails ([n – k]!)?
3. Suppose 1000 randomly selected residents of New York are asked to vote for Clinton and Trump. Assume that the residents of New York are equally divided on this issue. What is the binomial probability of 501 or more vote for Trump?
The moral of the lesson: the binomial probability refers to a set of n trials in which the probability P(k, n) that the number of “win” or successes will be obtained k times is P(k, n) = C(n, k) pkqn−k.
References:
1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.
2. Needham, J. (1959). Science and Civilization in China, vol. 3: Mathematics and the Sciences of the Heavens and the Earth. New York: Cambridge University Press.
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