Chapter 8: Probability Distributions

• What is a probability distribution?

  A list of all possible outcomes with their expected probabilities
• What is an example of a probability distribution when flipping a fair coin?

  The outcomes are heads or tails, each with a probability of $\frac{1}{2}$
• What is a binomial distribution?

  A type of probability distribution with only two possible outcomes
• What are the two possible outcomes when rolling a six on a dice in a binomial distribution?

  Success (landing on 6) or Failure (not landing on 6)
• How is a binomial distribution denoted?

  It is written as B(n, p)
• What does 'n' represent in the notation B(n, p)?

  The number of trials
• What does 'p' represent in the notation B(n, p)?

  The probability of success
• What is the first condition for a binomial distribution?

  There must be a fixed number of trials (n)
• What is the second condition for a binomial distribution?

  Each trial has 2 outcomes: success (p) or failure (q)
• What is the third condition for a binomial distribution?

  All trials are independent of each other
• What is the fourth condition for a binomial distribution?

  The probability of success is constant for every trial
• How can you determine if the binomial distribution is a suitable model for an event?

  You need to show if the event meets all four conditions
• How do you find probabilities using the binomial distribution?

  Use the formula $(p + q)^n$ to find the probabilities
• What is the first step in finding probabilities using the binomial distribution?

  Identify the two outcomes and their probabilities
• What is the second step in finding probabilities using the binomial distribution?

  Expand $(p + q)^n$ where n is the number of trials
• What is the third step in finding the probability of x successes?

  Find the term that has $p$ to the power of x successes
• What is the fourth step in finding the probability of x successes?

  Substitute the values of p and q into that term and calculate
• For the event of rolling a six on a dice, what are the values of p and q?

  p = $\frac{1}{6}$ and q = $\frac{5}{6}$
• What is the first method to find coefficients in a binomial distribution?

  Use Pascal’s Triangle
• What does Pascal’s Triangle start with?

  It starts with 1 in row 0
• How are the other numbers in Pascal’s Triangle found?

  By adding the two numbers directly above

  
• What is the expansion of $(p + q)^4$?


  $1p^4 +$$4p^3q^1 +$$6p^2q^2 +$$4p^1q^3 +$$1q^4$
• What happens to the power of p and q in the expansion of a binomial distribution?

  The power of p decreases by 1 and the power of q increases by 1
• What is the second method to find coefficients in a binomial distribution?

  Use the nCr button on a calculator
• What does n represent in the nCr notation?

  The number of trials
• What does r represent in the nCr notation?

  The number of successes
• How do you calculate the coefficient for 5 trials with 3 successes using nCr?

  Type ‘5’, ‘nCr’, ‘3’, ‘=’
• How do you find a range of probabilities in a binomial distribution?

  Calculate individual probabilities and then add them up
• How do you find the probability of ‘at least 1 success’?

  Calculate the probability of 0 successes and subtract from 1
• What is the mean (or expected value) of the binomial distribution B(n, p)?

  It is $np$
• For B(6, $\frac{1}{2}$), what is the mean?


  The mean is 3
• What shape is a normal distribution drawn as?

  A smooth, bell-shaped curve

  
• What is a common application of normal distribution?

  Modeling real-life situations like weights of apples or exam marks
• Where is most of the data in a normal distribution located?

  In the middle with similar values
• What effect does a larger standard deviation have on the normal distribution curve?

  It results in a lower curve
• What does the notation N(μ, σ²) represent in normal distribution?

  μ is the mean and σ² is the variance
• What is the first condition for a normal distribution?

  The data is continuous
• What is the second condition for a normal distribution?

  The distribution is symmetrical
• What is the third condition for a normal distribution?

  Mode, median, and mean are approximately equal
• When is normal distribution not suitable?

  When the data is skewed