Question

Suppose you toss a fair coin 10 times.

(a) Calculate the probability of getting at least 6 heads, using the exact distribution.

(b) Now repeat the calculate above, but approximate the probability using a normal random variable. Do your calculation both with and without the histogram correction. Which one is closer to the true answer?

Now suppose you toss a fair coin 1000 times.

(c) What is the probability of getting at least 520 heads? You can approximate this using a normal with the appropriate mean and variance. Do this with and without the histogram correction. (Does it matter if you use this correction?)

(d) What is the probability of getting at least 600 heads? Calculate this however you want.

Answer 1

Since coin is fair so getting a head in a single trial is P(H) = 0.50.

Let X is a random variable shows the number of heads out of 10. Here X has binomial distribution with parameters as follow:

n=10 and p= 0.5

(a)

The probability of getting at least 6 heads is

(b)

Using normal approximation, X has approximately normal distribution with mean and SD as follows:

Using without histogram correction:

The z-score for X = 6 is

Using z table, The probability of getting at least 6 heads is

Using with histogram correction:

The z-score for X = 6-0.5=5.5 is

The probability of getting at least 6 heads is

The calculations with histogram correction is closer to the true answer.

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Let X is a random variable shows the number of heads out of 10. Here X has binomial distribution with parameters as follow:

n=1000 and p= 0.5

Using normal approximation, X has approximately normal distribution with mean and SD as follows:

(c)

Using without histogram correction:

The z-score for X = 520 is

Using z table, The probability of getting at least 520 heads is

Using with histogram correction:

The z-score for X = 520-0.5 is

Using z table, The probability of getting at least 520 heads is

(d)

The z-score for X = 600-0.5 is

Using z table, The probability of getting at least 520 heads is