Slide 1

• Topic: Probability - Variance of Binomial Distribution

Slide 2

• Binomial Distribution:
  • A probability distribution that summarizes the number of successes in a fixed number of independent Bernoulli trials.
  • Each trial has two possible outcomes: success or failure.
  • The probability of success remains constant for each trial.

Slide 3

• Formula for Variance of Binomial Distribution:
  • Var(X) = np(1-p)
  • Where:
    • Var(X) is the variance of the binomial distribution
    • n is the number of trials
    • p is the probability of success in a single trial

Slide 4

• Example:
  • Consider flipping a fair coin 10 times.
  • The probability of getting heads (success) is 0.5.
  • The number of trials, n = 10.
  • Calculate the variance of the binomial distribution.

Slide 5

• Solution:
  • n = 10, p = 0.5
  • Variance of Binomial Distribution: Var(X) = np(1-p)
  • Var(X) = 10 * 0.5 * (1 - 0.5)
  • Var(X) = 10 * 0.5 * 0.5
  • Var(X) = 2.5

Slide 6

• Interpretation:
  • The variance of the binomial distribution for flipping a fair coin 10 times is 2.5.
  • This indicates that there is a variability in the number of heads obtained in repeated experiments of flipping the coin 10 times.

Slide 7

• Properties of Variance:
  • The variance is always non-negative.
  • If all trials have the same probability of success, the binomial distribution will have the highest variance when p = 0.5.

Slide 8

• Importance of Variance:
  • Variance provides a measure of the spread or dispersion of a probability distribution.
  • It helps to understand the variability in the outcomes of a binomial experiment.
  • Larger variances indicate greater spread or dispersion.

Slide 9

• Relationship between Variance and Standard Deviation:
  • The standard deviation is the square root of the variance.
  • It measures the average amount by which observations in the distribution differ from the mean.

Slide 10

• Example:
  • Find the standard deviation of the binomial distribution used in the previous example.

Slide 11

• Solution:
  • Variance of Binomial Distribution: Var(X) = np(1-p)
  • Var(X) = 10 * 0.5 * (1 - 0.5)
  • Var(X) = 10 * 0.5 * 0.5
  • Var(X) = 2.5
• Interpretation:
  • The variance of the binomial distribution for flipping a fair coin 10 times is 2.5.
  • This indicates that there is a variability in the number of heads obtained in repeated experiments of flipping the coin 10 times.
• Properties of Variance:
  • The variance is always non-negative.
  • If all trials have the same probability of success, the binomial distribution will have the highest variance when p = 0.5.
• Importance of Variance:
  • Variance provides a measure of the spread or dispersion of a probability distribution.
  • It helps to understand the variability in the outcomes of a binomial experiment.
  • Larger variances indicate greater spread or dispersion.
• Relationship between Variance and Standard Deviation:
  • The standard deviation is the square root of the variance.
  • It measures the average amount by which observations in the distribution differ from the mean.

Slide 12

• Example:
  • Consider shooting free throws in basketball.
  • The probability of making a free throw is 0.75.
  • A player shoots 20 free throws.
  • Calculate the variance of the binomial distribution.

Slide 13

• Solution:
  • n = 20, p = 0.75
  • Variance of Binomial Distribution: Var(X) = np(1-p)
  • Var(X) = 20 * 0.75 * (1 - 0.75)
  • Var(X) = 20 * 0.75 * 0.25
  • Var(X) = 3.75
• Interpretation:
  • The variance of the binomial distribution for shooting 20 free throws with a probability of success of 0.75 is 3.75.
  • This indicates that there is a variability in the number of successful free throws obtained in repeated experiments of shooting 20 free throws.

Slide 14

• Example:
  • Consider rolling a fair 6-sided die 50 times.
  • The probability of rolling a 3 (success) is 1/6.
  • Calculate the variance of the binomial distribution.

Slide 15

• Solution:
  • n = 50, p = 1/6
  • Variance of Binomial Distribution: Var(X) = np(1-p)
  • Var(X) = 50 * (1/6) * (1 - 1/6)
  • Var(X) = 50 * (1/6) * (5/6)
  • Var(X) = 4.63
• Interpretation:
  • The variance of the binomial distribution for rolling a fair 6-sided die 50 times with a probability of success of 1/6 is 4.63.
  • This indicates that there is a variability in the number of times a 3 is rolled in repeated experiments of rolling the die 50 times.

Slide 16

• Example:
  • Consider drawing a card from a standard deck of 52 cards 100 times.
  • The probability of drawing a spade (success) is 1/4.
  • Calculate the variance of the binomial distribution.

Slide 17

• Solution:
  • n = 100, p = 1/4
  • Variance of Binomial Distribution: Var(X) = np(1-p)
  • Var(X) = 100 * (1/4) * (1 - 1/4)
  • Var(X) = 100 * (1/4) * (3/4)
  • Var(X) = 18.75
• Interpretation:
  • The variance of the binomial distribution for drawing a card from a standard deck of 52 cards 100 times with a probability of success of 1/4 is 18.75.
  • This indicates that there is a variability in the number of times a spade is drawn in repeated experiments of drawing a card 100 times.

Slide 18

• Example:
  • Consider a genetics experiment where the probability of a certain trait being present (success) in a population is 0.2.
  • The experiment is conducted on 500 individuals.
  • Calculate the variance of the binomial distribution.

Slide 19

• Solution:
  • n = 500, p = 0.2
  • Variance of Binomial Distribution: Var(X) = np(1-p)
  • Var(X) = 500 * 0.2 * (1 - 0.2)
  • Var(X) = 500 * 0.2 * 0.8
  • Var(X) = 80
• Interpretation:
  • The variance of the binomial distribution for a genetics experiment conducted on 500 individuals with a probability of success of 0.2 is 80.
  • This indicates that there is a variability in the number of individuals with the certain trait in repeated experiments.

Slide 20

• Summary:
  • Variance of Binomial Distribution: Var(X) = np(1-p)
  • The variance provides a measure of the spread or dispersion in the outcomes of a binomial experiment.
  • Larger variances indicate greater spread or dispersion.
  • The standard deviation is the square root of the variance. It measures the average amount by which observations differ from the mean.

Slide 21

• Properties of Binomial Distribution:
  • It is a discrete probability distribution.
  • It is symmetric when p = 0.5.
  • It is right-skewed when p < 0.5 and left-skewed when p > 0.5.
  • The mean of the binomial distribution is equal to np.
  • The mode of the distribution is the value of x corresponding to the highest probability.

Slide 22

• Example:
  • Consider rolling a fair 6-sided die 12 times.
  • The probability of rolling a 6 (success) is 1/6.
  • Calculate the mean and mode of the binomial distribution.

Slide 23

• Solution:
  • n = 12, p = 1/6
  • Mean = np = 12 * (1/6) = 2
  • Mode = floor((n+1)p) = floor((12+1)(1/6)) = floor(2.166) = 2
• Interpretation:
  • The mean of the binomial distribution for rolling a fair 6-sided die 12 times with a probability of success of 1/6 is 2.
  • The mode is also 2, which indicates that rolling a 6 is the most probable outcome.

Slide 24

• Example:
  • Consider drawing a card from a standard deck of 52 cards 20 times.
  • The probability of drawing a heart (success) is 1/4.
  • Calculate the mean and mode of the binomial distribution.

Slide 25

• Solution:
  • n = 20, p = 1/4
  • Mean = np = 20 * (1/4) = 5
  • Mode = floor((n+1)p) = floor((20+1)(1/4)) = floor(5.25) = 5
• Interpretation:
  • The mean of the binomial distribution for drawing a card from a standard deck of 52 cards 20 times with a probability of success of 1/4 is 5.
  • The mode is also 5, which indicates that drawing a heart is the most probable outcome.

Slide 26

• Example:
  • Consider conducting an experiment where the probability of a certain event happening (success) is 0.3.
  • The experiment is repeated 100 times.
  • Calculate the mean and mode of the binomial distribution.

Slide 27

• Solution:
  • n = 100, p = 0.3
  • Mean = np = 100 * 0.3 = 30
  • Mode = floor((n+1)p) = floor((100+1)(0.3)) = floor(30.3) = 30
• Interpretation:
  • The mean of the binomial distribution for an experiment with a probability of success of 0.3, repeated 100 times, is 30.
  • The mode is also 30, indicating that the event occurring 30 times is the most probable outcome.

Slide 28

• Binomial Distribution and Normal Distribution:
  • For large values of n, a binomial distribution can be approximated by a normal distribution.
  • The mean of the normal distribution is equal to np.
  • The standard deviation of the normal distribution is equal to √(np(1-p)).

Slide 29

• Example:
  • Consider flipping a fair coin 1000 times.
  • The probability of getting heads (success) is 0.5.
  • Approximate the binomial distribution with a normal distribution.

Slide 30

• Solution:
  • n = 1000, p = 0.5
  • Mean = np = 1000 * 0.5 = 500
  • Standard Deviation = √(np(1-p)) = √(1000 * 0.5 * (1 - 0.5)) = √(1000 * 0.5 * 0.5) = √250 = 15.81
• Interpretation:
  • The mean of the approximated normal distribution for flipping a fair coin 1000 times is 500.
  • The standard deviation is 15.81, indicating the spread of the distribution.
  • This approximation is useful for large sample sizes, where the binomial distribution becomes similar to a normal distribution.