Probability Distributions: Binomial and Normal Distributions
What Is a Probability Distribution?
A probability distribution describes how the probabilities are distributed over the values of a random variable. There are two main types of distributions that are particularly useful in A-Level Mathematics: the binomial distribution and the normal distribution.
Binomial Distribution
The binomial distribution describes the number of successes in a fixed number of independent trials, where each trial has two possible outcomes (success or failure). It is used when the probability of success is the same in each trial.
The probability of obtaining exactly \( k \) successes in \( n \) trials is given by the formula:
\[
P(X = k) = \binom{n}{k} p^k (1 – p)^{n – k}
\]
Where:
- \( n \) is the number of trials,
- \( k \) is the number of successes,
- \( p \) is the probability of success on a single trial,
- \( \binom{n}{k} \) is the binomial coefficient.
Properties of the Binomial Distribution
- Mean: \( \mu = np \)
- Variance: \( \sigma^2 = np(1 – p) \)
Normal Distribution
The normal distribution is a continuous probability distribution that is symmetric about the mean. It is often referred to as a “bell curve” because of its shape. The normal distribution is characterised by the mean (\( \mu \)) and the standard deviation (\( \sigma \)).
The probability density function for a normal distribution is given by:
\[
f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x – \mu)^2}{2\sigma^2}}
\]
Properties of the Normal Distribution
- The mean, median, and mode of a normal distribution are all equal.
- The total area under the curve is equal to 1.
- About 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations.
Applications of Binomial and Normal Distributions
Binomial Distribution in Real Life
- Coin Tossing: The binomial distribution is used to model the number of heads in a series of coin tosses.
- Quality Control: In manufacturing, it can model the number of defective items in a batch of products.
Normal Distribution in Real Life
- Height and Weight: Many natural characteristics, such as human height and weight, follow a normal distribution.
- Examination Scores: Examination scores in large groups often follow a normal distribution.
Example Problem
Problem: A coin is tossed 10 times. What is the probability of getting exactly 6 heads if the probability of heads on each toss is 0.5?
Solution:
Using the binomial distribution formula:
\[
P(X = 6) = \binom{10}{6} (0.5)^6 (0.5)^4 = \binom{10}{6} (0.5)^{10}
\]
First, calculate the binomial coefficient:
\[
\binom{10}{6} = \frac{10!}{6!(10 – 6)!} = 210
\]
Now calculate the probability:
\[
P(X = 6) = 210 \times (0.5)^{10} = 210 \times \frac{1}{1024} = 0.205
\]
So, the probability of getting exactly 6 heads is 0.205.
Common Mistakes in Probability Distributions
- Misunderstanding Binomial Conditions: Ensure the trials are independent and that the probability of success is constant.
- Incorrect Use of the Normal Distribution: The normal distribution should be used for continuous data and may require the use of z-scores when calculating probabilities for specific values.
Practice Questions
- A bag contains 5 red and 3 green marbles. If two marbles are selected, what is the probability that both are red?
- A dataset follows a normal distribution with a mean of 50 and a standard deviation of 5. What is the probability that a randomly selected value is between 45 and 55?
- In a factory, 80% of products pass the quality control test. If 10 products are tested, what is the probability that exactly 8 pass the test?