Using the normal approximation to the binomial distribution,...

Using the normal approximation to the binomial distribution, and tables [or calculator for $\phi(t)$ ], find the approximate probability of each of the following:
Between 499,000 and 501,000 heads in $10^{6}$ tosses of a coin.

Key Concepts

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. It is a discrete probability distribution that is used to represent problems where outcomes are binary, such as heads or tails in a coin toss.

Normal Approximation to the Binomial Distribution

For a large number of trials, the binomial distribution can be approximated by a normal distribution with mean np and variance np(1-p). This approximation simplifies calculations, especially when dealing with large sample sizes, and is valid under conditions where the sample size is sufficiently large and the probability of success is not too close to 0 or 1.

Continuity Correction

When using a continuous distribution, like the normal distribution, to approximate a discrete distribution, a continuity correction is applied. This involves adjusting the intervals used in the calculation (by adding or subtracting 0.5) to capture the area corresponding more accurately to the discrete probability mass.

Central Limit Theorem

The Central Limit Theorem states that, given a large enough sample size, the sum or average of independent and identically distributed random variables will be approximately normally distributed, regardless of the original distribution. This theorem underpins the use of the normal approximation for the binomial distribution when the number of trials is large.

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