Probability Theory Quiz

Probability Theory Quiz

1. A coin is flipped three times. What is the probability of getting exactly two heads? Assuming the coin is fair.

2. A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. If a marble is drawn randomly, what is the probability it is either red or blue?

3. A company produces light bulbs. The probability that a bulb is defective is 0.05. If a sample of 10 bulbs is selected, what is the variance in the number of defective bulbs?

4. A game involves rolling a six-sided die. If you win 5ifyourolla6,andlose5 if you roll a 6, and lose 5ifyourolla6,andlose2 if you roll any other number, what is the expected value of this game?

5. A student guesses randomly on a multiple-choice exam with 5 questions. Each question has 4 possible answers. What is the probability of getting exactly 3 questions correct?

Study Notes: Probability Theory, Variance, Chance Events & Expectation

• Probability Theory:

  • Probability measures the likelihood of an event occurring. It ranges from 0 (impossible) to 1 (certain).
  • The sum of probabilities of all mutually exclusive outcomes in a sample space equals 1.
  • Conditional probability describes the probability of an event given that another event has already occurred.
  • Independent events have no influence on each other's probability.

• Chance Events:

  • Chance events are events where the outcome is not predictable with certainty.
  • Probability quantifies the likelihood of chance events.
  • Examples include flipping a coin, rolling dice, or drawing a card.

• Variance:

  • Variance is a measure of the spread or dispersion of a probability distribution. High variance indicates greater variability in outcomes.
  • Variance is calculated as the average of the squared deviations from the mean.
  • Variance is helpful for understanding the risk associated with uncertain outcomes.

• Expectation of a Random Variable:

  • The expected value (expectation) of a random variable is the average outcome if the process repeats many times. It is calculated as the sum of each possible outcome multiplied by its probability.
  • The expectation is a central measure that summarizes a random variable's possible outcomes.
  • The expected value might not be a possible outcome in a single trial.

Key Formulas

• Probability of an event: P(A) = number of favorable outcomes / total number of outcomes
• Probability of complementary events: P(not A) = 1 - P(A), where A is the event in question.
• Variance calculation: Variance(X) = E[(X - μ)2] where μ (mu) is the mean (expected value).
• Expected value: E(X) = Σ [ xi * P(xi) ] where xi are possible outcomes and P(xi) are their respective probabilities

Example Calculations (Illustrative)

• Example 1: (Coin flips) To calculate the probability of getting exactly 2 heads in 3 coin flips, we list all possible outcomes (HHH, HHT, HTH, THH, HTT, THT, TTH, TTT).
  • There are 8 total outcomes.
  • The outcomes with exactly two heads are HHT, HTH, THH: 3 outcomes.
  • P(exactly 2 heads) = 3 / 8 = 0.375
• Example 2: (Variance of defective bulbs).
  • To find the variance, you would first calculate the mean number of defects; then calculate the squared deviations from that mean. The mean for a binomial distribution is mean = np, where n = number of trials and p = probability of success.
• Example 3 (expected value of the game):
  • List all possible outcomes of the die roll.
  • Multiply each payout by the probability of that outcome.
  • Sum these results.
• Example 4 (multiple-choice exam):
  • Calculate the probability of getting one correct answer.
  • Use the binomial probability formula (nCr * px * (1-p)n-x) to calculate the probability of getting exactly 3 correct given 5 questions with a 1/4 chance of success in each if you guess randomly.

Test your understanding of probability theory with this quiz that covers key concepts such as chance events, variance, and expected value. From flipping coins to drawing marbles, each question will challenge your knowledge and problem-solving skills in probability.