CLASS 10

Factorials

• Definition: Factorial of a non-negative integer n, denoted n!, represents the product of all positive integers less than or equal to n.

  • Formula:
    n! = n × (n - 1) × (n - 2) × … × 2 × 1.

  • Convention: 0! = 1.

Combinations

• Notations: Denoted as (n k) or nCk, represents the number of ways to choose k items from n items where the order of selection does not matter.

  • Formula:
    nCk = n! / ((n - k)! × k!)

• Special Combinations:

  • (n n) = 1

  • (n 0) = 1

  • (n 1) = n

  • (n n - 1) = n

Chapter 13: Binomial Distribution

• Definition: A probability distribution is categorized as binomial if it meets the following criteria:

  1. There is a fixed number of trials, denoted as n.

  2. The trials are independent of each other.

  3. There are only two outcomes for each trial: success and failure.

  4. The probability of success, denoted as p, remains constant across trials.

Examples of Binomial Distributions

• Determine if the following scenarios represent binomial distributions: a) Tossing a weighted coin 10 times to see how many tails occur.

  • n = 10 (trials), p = probability of tails.
    b) Rolling a die until a 6 appears.

  • Not a binomial distribution (variable number of trials).
    c) Asking 200 people their age (X is their age).

  • Not a binomial distribution (continuous variable).

Binomial Probability Formula

• Formula: P(X = x) = (n x) × p^x × q^(n−x) where:

  • n = number of trials

  • x = number of successes among n trials

  • p = probability of success in one trial

  • q = probability of failure in one trial (q = 1 - p)

• P(X = x) represents the probability of getting exactly x successes in n trials.

Example Calculation

• Assumption: Probability that the average resting heart rate is greater than 80 beats per minute is 68%.

• Given a random sample of 10 adult females:
  a) Identify if it's a binomial distribution. n = 10, p = 0.68.
  b) Calculate P(X = 0) and P(X = 10).
  c) Find the probability that at least 2 selected females have a resting heart rate > 80 bpm.

Statistics for Binomial Distribution

• Mean:
  μ = n × p

• Variance:
  σ² = n × p × q

• Standard Deviation:
  σ = √(n × p × q)

• Example: Given p = 0.68 and n = 10:
  a) Find mean and standard deviation.
  b) Determine unusual values in the context of this scenario.

Normal Approximation to Binomial Distribution

• Condition: If n × p ≥ 10 and n × q ≥ 10, the binomial distribution can be approximated by a normal distribution, denoted N(μ, σ), where:

  • μ = n × p

  • σ = √(n × p × q)

• Important Note: If either n × p < 10 or n × q < 10, do not use the normal approximation.

Continuity Correction

• Concept: When using a normal approximation for a binomial distribution, adjust the discrete whole number x by adding or subtracting 0.5.

  • Use either x - 0.5 or x + 0.5 to account for the continuous nature of the normal distribution.

• Example: Given that approximately 17% of households in Canada have a pet dog, find the probability that at least 19 out of 90 randomly selected households have a pet dog.

Continuity Correction Factor Table

• Binomial (discrete) to Normal (continuous) relationships:

  • P(X = n) ⟶ P(n - 0.5 < X < n + 0.5)

  • P(X > n) ⟶ P(X > n + 0.5)

  • P(X ≥ n) ⟶ P(X > n - 0.5)

  • P(X < n) ⟶ P(X < n - 0.5)

  • P(X ≤ n) ⟶ P(X < n + 0.5)

• Reason for Using Continuity Correction: The binomial distribution is discrete, while the normal distribution is continuous. The correction allows for a better approximation.

Practice Questions

1. Find the number of ways to choose 3 cards from a deck of 52.

   • Methods include choosing all at once, one at a time with replacement, and one at a time without replacement.

2. Calculate factorials:

   • 3! = 6

   • 5! = 120

   • 7! = 5040

3. Identify binomial distributions from various scenarios, providing n and p.
   a) Surveying 100 people about news viewership.
   b) Rolling a die 100 times.
   c) Drawing cards for a poker hand.
   d) Rolling a coin until a head appears.
   e) Rolling 6 dice and counting sixes.
   … (additional knowledge questions continuing as appropriate)