IB DP Maths: AA HL复习笔记4.5.1 The Binomial Distribution

What is a binomial distribution? 

• A binomial distribution is a discrete probability distribution

What are the important properties of a binomial distribution?

• The expected number (mean) of successful trials is

• • You are given this in the formula booklet
  • Square root to get the standard deviation
• The distribution can be represented visually using a vertical line graph
  • If p is close to 0 then the graph has a tail to the right
  • If p is close to 1 then the graph has a tail to the left
  • If p is close to 0.5 then the graph is roughly symmetrical
  • If p = 0.5 then the graph is symmetrical

How do I set up a binomial model?

• Identify what a trial is in the scenario
  • For example: rolling a dice, flipping a coin, checking hair colour
• Identify what the successful outcome is in the scenario
  • For example: rolling a 6, landing on tails, having black hair
• Identify the parameters
  • n is the number of trials and p is the probability of success in each trial
• Make sure you clearly state what your random variable is
  • For example, let be the number of students in a class of 30 with black hair

What can be modelled using a binomial distribution?

• Anything that satisfies the four conditions
• What can not be modelled using a binomial distribution?

• Anything where the number of trials is not fixed or is infinite
  • The number of emails received in an hour
  • The number of times a coin is flipped until it lands on heads
• Anything where the outcome of one trial affects the outcome of the other trials
  • The number of caramels that a person eats when they eat 5 sweets from a bag containing 6 caramels and 4 marshmallows
    • If you eat a caramel for your first sweet then there are less caramels left in the bag when you choose your second sweet
  • Anything where there are more than two outcomes of a trial
    • A person's shoe size
    • The number a dice lands on when rolled
  • Anything where the probability of success changes
    • The number of times that a person can swim a length of a swimming pool in under a minute when swimming 50 lengths
      • The probability of swimming a lap in under a minute will decrease as the person gets tired
      • The probability is not constant