Binomial and Normal Distributions

Probability and statistics are important branches of mathematics that help us understand the likelihood of events and the various outcomes to expect in everyday life. Of the various probability distributions used in these fields, the binomial distribution and the normal distribution are two fundamental models that describe different types of data. It is important for grade 12 math students to understand these distributions, as they are widely applicable in fields ranging from science and engineering to finance and social sciences.

Binomial distribution

The binomial distribution is a discrete probability distribution. It models the number of successes in a fixed number of trials of a binary experiment. A binary experiment is one where there are only two possible outcomes, usually referred to as “success” and “failure”. Let us learn more about the binomial distribution with the help of definition, formula, and examples.

Basic definition

The binomial distribution is defined by the following characteristics:

• Number of trials (n): The number of independent experiments or trials.
• Probability of success (p): The probability of achieving success in a single experiment.
• Probability of failure (q or 1-p): The probability of failure in a single experiment.
• Number of successes (k): The typical number of successes in n trials.

The probability of getting exactly k successes in n trials is given by the formula:

    P(x=k) = C(n,k) * P^k * (1-P)^(n-k)

where P(X = k) is the probability of getting k successes, and C(n, k) is the combination function:

    C(n, k) = n! / [k!(n-k)!]

Example of binomial distribution

Consider a simple example: tossing a coin 5 times. What is the probability of getting heads exactly 3 times?

Here, we have the following parameters:

• n = 5 (number of trials)
• p = 0.5 (the probability of getting heads in a trial, since a fair coin has two sides)
• k = 3 (number of successes, in this case, heads)

Substitute these values into the binomial formula:

    P(x = 3) = C(5, 3) * (0.5)^3 * (1-0.5)^(5-3)
             = 10 * 0.125 * 0.25
             = 0.3125

So, the probability of getting heads exactly 3 times when a coin is tossed 5 times is 0.3125 or 31.25%.

Visual representation

This SVG shows a bar graph of the binomial distribution for tossing a coin 5 times (n=5), with the probability of success (heads) being p=0.5.

Normal distribution

The normal distribution is a continuous probability distribution that is symmetric around the mean, indicating that data near the mean occur more often than data away from the mean. As a graph, the normal distribution will appear as a bell curve.

Properties of normal distribution

The normal distribution is defined by its mean (μ) and standard deviation (σ). Its characteristic properties are:

• The total area under the curve is 1.
• The curve is symmetric around the mean (μ).
• About 68% of the data falls within one standard deviation (σ) of the mean.
• Approximately 95% of the data falls within two standard deviations (2σ) of the mean.
• Approximately 99.7% of the data falls within three standard deviations (3σ) of the mean.

Normal distribution formula

The probability density function (PDF) of the normal distribution is described by the formula:

    f(x) = (1 / (σ * sqrt(2π))) * exp(-0.5 * ((x - μ) / σ)^2)

Example of normal distribution

Suppose the test scores of a class are normally distributed, with a mean (μ) of 70 and a standard deviation (σ) of 10. You want to find the probability that a randomly selected student scored between 60 and 80.

To find this, you would calculate the area under the normal curve between these two values.

Visual representation

This SVG presents a basic diagram of a normal distribution curve, displaying the standard deviation from the mean.

Z-score

In the normal distribution, the Z-score is used to determine how far and in which direction a data point is from the mean. The Z-score is calculated using the formula:

    Z = (X − μ) / σ

where X is the value, μ is the mean of the distribution, and σ is the standard deviation.

Example of Z-score calculation

Suppose we have an example of a previous exam with a mean of 70 and a standard deviation of 10. A student scored 85 marks. What is the Z-score?

Substitute the values into the Z-score formula:

    Z = (85 - 70) / 10 = 1.5

This means that a score of 85 is 1.5 standard deviations above the mean.

Relationship between binomial and normal distribution

Under certain conditions, the binomial distribution can be approximated by a normal distribution. This happens when:

• The number of trials (n) is large.
• The probability of success (p) is neither very close to 0 nor to 1.

The general rule is that the normal approximation is considered appropriate when both np and n(1-p) are greater than 5. In such cases, the mean (μ) and standard deviation (σ) for the normal distribution are given by:

    μ = np
    σ = sqrt(np(1-p))

Example of normal approximation

Imagine you have a heterogeneous data set of 1000 coin tosses, where each toss is independent. The event is a "heads" toss with a probability of 0.5.

We apply the normal approximation:

• μ = np = 1000 * 0.5 = 500
• σ = sqrt(np(1-p)) = sqrt(1000 * 0.5 * 0.5) = sqrt(250) = 15.81

If we wanted to find the probability of heads coming up between 490 and 510, we would look at the Z-score and find the area under the normal curve between these two values.

Conclusion

The binomial and normal distributions are fundamental concepts in probability and statistics. They provide insight into a wide variety of phenomena, providing tools for understanding the behavior of discrete and continuous random variables. From simple experiments like coin tosses to more complex real-world scenarios, these distributions help us make informed decisions based on probability models. The ability to use these tools not only enriches mathematical understanding but also enhances analytical skills needed in a variety of fields. Understanding and using these distributions effectively requires mastering their properties, formulas, and real-world applications, making them indispensable in both academic and professional contexts.

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