Question-69
Six fair coin are tossed 6400 times. Using the Poisson distribution, find the approximate probability of getiing six heads \(r\) times.
Use the concept of “Poisson distribution is a limiting case of the binomial distribution under the following conditions:
(i). \(n\), the number of trials is indefinitely large, i.e., \(n \to \infty\).
(ii). \(p\), the constant probability of success for each trial is indefinitely small, i.e., \(p \to 0\).
(iii). \(np = \lambda,\) (say) is finite.
The probability of obtaining six heads in one throw of six coins (a single trial), is \(p = \left(\dfrac{1}{2}\right)^6\).
\(\lambda = np = 6400 \times \left(\dfrac{1}{2}\right)^6 = 100\)
Hence, using Poisson Probability distribution, the required probability of getting 6 heads \(r\) times is given by:
\(P(X = r) = \dfrac{e^{-\lambda} \lambda^r}{r!} = \dfrac{e^{-100} (100)^r}{r!} ~;~ r = 0, 1, 2, \ldots\)