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A fair coin is tossed 10 times. Find the probability of getting at least 4 heads and at most 6 heads.

Let X be the probability distribution of getting x heads. We need to find $k$ such that $k = P(4 \le X \le 6)$.

It is a binomial distribution with n = 10, p = q = 0.5. Then, k = B(6, 10, 0.5) – B( 3 , 10, 0.5) = 0.656 (from table). This is because, for this discrete distribution, we have to subtract the values of P(X = 0, 1, 2, 3) from B(6, 10, 0.5).

When I approximate it by normal distribution N(5, 2.5), I found that I have to use $X_1 = 3$ (instead of 4) and $X_2 = 6$ to get the respective standard scores $Z_1 = –1.265$ and $Z_2 = 0.633$. The corresponding area under the standard normal curve is then 0.634 giving a very close approximate to the result previously found.

My question is:- using $X_1 = 3$ and $X_2 = 6$ to find the corresponding area under the normal curve will include those X = 3.xxxxx ~ X = 4.0. Shouldn't those areas be excluded according to the question?

Mick
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1 Answers1

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When approximating a discrete variable with a continuous one (asymptotics) the best approach is to "spread" the discrete variable half-way up and down, i.e. to approximate the pmf $y=p(X=x)$ with the step wise continuous pdf$y=p(x-1/2 \le X < x+1/2)$ and consequently for the CDF.

In your case $P(4 \le X \le 6) \; \Rightarrow N (3.5 \le X < 6.5)$

G Cab
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  • Thanks for the info. The book did mention the 0.5 part but it didn't say anything about it. Instead, it says --- for simplicity, we will drop the o.5 part. – Mick Aug 10 '20 at 17:02
  • "for simplicity, we will drop the $0.5$ part" .. what to say ? that it depends on the accuracy you want (and on the width/st dev. ) – G Cab Aug 10 '20 at 17:20
  • Clarification: The book did mention that one can use the limit theorem by De Moivre and Laplace to approximate a B-distribution by a N-distribution. In which, there is a correction factor of $\pm 0.5$. Before this question, I don’t know what are they for and how to use them. Later on, in another chapter on finding the critical value of a hypothesis, the author in an illustrative example mentioned “For the sake of simplicity we shall disregard the term 0.5 in …”. – Mick Aug 11 '20 at 04:06