Unit 15: Probability and Statistics
Counting outcomes, weighing chance events, and describing how a whole batch of data spreads out around its center.
9 topics · 35 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
The Fundamental Counting Principle
- How it worksHow does multiplying the number of choices at each stage count every possible outcome without listing them?
- Worked exampleFind how many different license plates are possible with 3 letters followed by 4 digits.
- Common mistakeWhy doesn't the counting principle add the number of choices at each stage instead of multiplying them?
Your turn1 of 2
Find how many different license plates are possible with 4 letters followed by 4 digits.Permutations
- How it worksHow does the permutation formula account for having fewer choices left at each step?
- Worked exampleFind how many ways 4 runners can finish first, second, and third in a race of 8 runners.
- Common mistakeWhy isn't a permutation the same thing as a combination?
- Going furtherWhat happens to a permutation count when some of the items being arranged are actually identical to each other?
Your turn1 of 2
Find how many ways 3 runners can finish first, second, and third, out of a race of 15 runners.Combinations
- How it worksHow does dividing a permutation by the number of ways to reorder the chosen group produce a combination?
- Worked exampleFind how many different 5 person committees can be formed from a group of 12 people.
- Common mistakeWhy does order not matter in a combination the way it matters in a permutation?
- Ties togetherHow does a combination come from a permutation once you divide out the orderings that no longer matter?
Your turn1 of 2
Find how many different 5 person committees can be formed from a group of 24 people.Probability of Compound Events
- How it worksHow does multiplying two probabilities together find the chance that both independent events occur?
- Worked exampleFind the probability of rolling a 6 on a die and then flipping heads on a coin.
- Common mistakeWhy can't the probability of A or B always be found by just adding the probability of A and the probability of B?
Your turn1 of 2
Find the probability of rolling a specific number on a fair 15-sided die and then flipping heads on a fair coin.Conditional Probability
- How it worksHow does dividing by the probability of the known event narrow the sample space down to a conditional probability?
- Worked exampleFind the probability that a card drawn from a deck is a king, given that it is known to be a face card.
- Common mistakeWhy isn't the probability of A given B the same as the probability of B given A?
- Going furtherWhat happens to a conditional probability calculation when the two events turn out to be independent?
Your turn1 of 2
Given that a card drawn from a group of 17 face cards is known to be a face card, and 4 of those face cards are kings, find the probability that it is a king.The Binomial Theorem
- How it worksHow do combinations generate every coefficient in the expansion of a binomial raised to a power?
- Worked exampleFind the expansion of (x + 2)4 using the binomial theorem.
- Common mistakeWhy doesn't (x + y)n expand into just xn plus yn?
- Going furtherWhat happens to the signs in a binomial expansion when you expand (x - y) to a power instead of (x + y)?
- Ties togetherHow do the coefficients in the binomial theorem come directly from n choose k combinations?
Your turn1 of 2
Find the coefficient of x2 in the expansion of (x + 7)4.Binomial Probability
- How it worksHow does the binomial probability formula combine combinations with repeated multiplication to count successes?
- Worked exampleFind the probability of getting exactly 3 heads in 5 flips of a fair coin.
- Common mistakeWhy doesn't the most likely outcome of many coin flips have to be exactly half heads and half tails?
- Ties togetherHow does binomial probability combine the combination count from the binomial theorem with multiplying independent probabilities?
Your turn1 of 2
Find the probability of getting exactly 2 heads in 8 flips of a fair coin.Mean, Variance, and Standard Deviation of a Data Set
- How it worksHow does squaring each deviation from the mean before averaging them produce the variance of a data set?
- Worked exampleFind the mean and standard deviation of the data set 4, 8, 6, 5, 7.
- Common mistakeWhy doesn't a standard deviation of zero mean something went wrong with the data?
Your turn1 of 2
Find the mean of the data set 14, 13, 19, 6, 2.The Normal Distribution
- How it worksHow do the mean and standard deviation of a normal distribution control its center and its spread?
- Worked exampleFind the percent of data falling within 2 standard deviations of the mean using the 68-95-99.7 rule.
- Common mistakeWhy doesn't every roughly symmetric, bell-shaped set of data automatically count as normally distributed?
- Going furtherWhat happens to the 68-95-99.7 percentages when a data set is only roughly, not perfectly, bell-shaped?
- Ties togetherHow does the normal distribution's spread connect to the standard deviation of the underlying data set?
Your turn1 of 2
A normal distribution has mean 73 and standard deviation 13. Using the 68-95-99.7 rule, find the upper bound of the range containing 95% of the data (within 2 standard deviations of the mean).