Tree diagrams

• What are the two possible outcomes for Sam's first game?

  Win or lose
• What is the probability of winning on the spinner?

  $\frac{2}{5}$
• What is the probability of losing on the spinner?

  $\frac{3}{5}$
• How do you represent the outcomes of two consecutive games using a probability tree diagram?

  Draw two branches for the first game and two branches for each outcome of the second game
• What is the probability of Sam winning both games?

  $\frac{2}{5} \times \frac{2}{5} =$$\frac{4}{25}$
• What is the probability of rolling a six on a fair six-sided dice?

  $\frac{1}{6}$
• What is the probability of not rolling a six on a fair six-sided dice?

  $\frac{5}{6}$
• What is the probability of Ria winning a prize (rolling a six and then throwing tails)?

  $\frac{1}{6} \times \frac{1}{2} =$$\frac{1}{12}$
• What is the probability of selecting a green counter from a bag containing 3 green and 5 red counters?

  $\frac{3}{8}$
• What is the probability of selecting a red counter from a bag containing 3 green and 5 red counters?

  $\frac{5}{8}$
• What is the probability of selecting two red counters in a row with replacement?

  $\frac{5}{8} \times \frac{5}{8} =$$\frac{25}{64}$
• What is the probability of selecting one red counter in two picks with replacement?

  $\frac{15}{64} +$$\frac{15}{64} =$$\frac{30}{64}$
• What is the probability of selecting two counters of the same color with replacement?

  $\frac{9}{64} +$$\frac{25}{64} =$$\frac{34}{64}$
• What is the probability that Luke passes the math test?

  0.8
• What is the probability that Luke passes the English test?

  0.7
• What are the steps to complete a probability tree diagram?

  1. Identify all possible outcomes for each event
  2. Draw branches for each outcome
  3. Label each branch with the corresponding probability
  4. Multiply probabilities along the branches to find combined probabilities
• What are the key differences between independent and dependent events in probability?

  Independent events:

  • The outcome of one event does not affect the other
  • Probabilities are multiplied directly

  
  

  Dependent events:

  • The outcome of one event affects the other
  • Probabilities change based on previous outcomes
• How do you calculate the probability of multiple independent events occurring?

  Multiply the probabilities of each individual event
• How do you calculate the probability of at least one of multiple independent events occurring?

  1 - (Probability of none of the events occurring)
• What are the common mistakes when working with probability tree diagrams?

  • Adding probabilities instead of multiplying along branches
  • Forgetting to consider all possible routes
  • Not labeling branches with correct probabilities
  • Confusing independent and dependent events
• What is the difference between mutually exclusive and independent events?

  Mutually exclusive events:

  • Cannot occur at the same time
  • P(A and B) = 0

  
  

  Independent events:

  • Occurrence of one does not affect the other
  • P(A and B) = P(A) * P(B)
• How do you calculate the probability of either of two mutually exclusive events occurring?

  Add the probabilities of each event
• How do you calculate the probability of either of two non-mutually exclusive events occurring?

  P(A) + P(B) - P(A and B)
• How do you convert a probability from a fraction to a decimal?

  Divide the numerator by the denominator
• What is the difference between probability and odds?

  Probability:

  • Ratio of favorable outcomes to total possible outcomes

  
  

  Odds:

  • Ratio of favorable outcomes to unfavorable outcomes
• What is the difference between theoretical and experimental probability?

  Theoretical probability:

  • Based on possible outcomes
  • Calculated mathematically

  
  

  Experimental probability:

  • Based on actual trials
  • Calculated from observed results
• How do you convert a probability from a decimal to a percentage?

  Multiply by 100
• How do you calculate the odds in favor of an event?

  Number of favorable outcomes : Number of unfavorable outcomes
• How do you calculate the odds against an event?

  Number of unfavorable outcomes : Number of favorable outcomes
• What is the difference between permutations and combinations?

  Permutations:

  • Order matters
  • nPr = n! / (n - r)!

  
  

  Combinations:

  • Order doesn't matter
  • nCr = n! / (r! * (n - r)!)
• How do you calculate the number of permutations of n objects taken r at a time?

  nPr = n! / (n - r)!
• How do you calculate the number of combinations of n objects taken r at a time?

  nCr = n! / (r! * (n - r)!)
• What is the difference between discrete and continuous probability distributions?

  Discrete:

  • Finite or countably infinite outcomes
  • Probability mass function

  
  

  Continuous:

  • Infinite possible outcomes
  • Probability density function
• What are the properties of a probability distribution?

  • All probabilities are between 0 and 1
  • Sum of all probabilities equals 1
• What is the difference between a binomial and a normal distribution?

  Binomial:

  • Discrete
  • Fixed number of trials
  • Two possible outcomes

  
  

  Normal:

  • Continuous
  • Bell-shaped curve
  • Infinite possible outcomes
• How do you calculate the expected value of a discrete probability distribution?

  Sum of (each outcome * its probability)
• How do you calculate the variance of a discrete probability distribution?

  Sum of ((each outcome - mean)^2 * its probability)
• How do you calculate the standard deviation of a discrete probability distribution?

  Square root of the variance
• What is the difference between a population and a sample in probability?

  Population:

  • Entire group of interest
  • Parameters describe it

  
  

  Sample:

  • Subset of the population
  • Statistics describe it
• What is the difference between a parameter and a statistic?

  Parameter:

  • Describes a population
  • Usually unknown

  
  

  Statistic:

  • Describes a sample
  • Used to estimate parameters