Review for final exam : probability unit
MIT 18.05 Spring 2022
Sets and counting
• Sets: ∅, union, intersection, complement Venn diagrams, products
• Counting: inclusion-exclusion, rule of product,
= (𝑛
permutations 𝑛𝑃𝑘, combinations 𝑛𝐶𝑘 𝑘)
Problem 1. Consider the nucleotides 𝐴, 𝐺, 𝐶, 𝑇 .
(a) How many ways are there to make a sequence of 5 nucleotides.
(b) How many sequences of length 5 are there where no adjacent nucleotides are the same
(c) How many sequences of length 5 have exactly one 𝐴?
Problem 2. (a) How many 5 card poker hands are there?
(b) How many ways are there to get a full house (3 of one rank and 2 of another)?
(c) What’s the probability of getting a full house?
Problem 3. (Counting)
(a) How many arrangements of the letters in the word probability are there?
(b) Suppose all of these arrangements are written in a list and one is chosen at random.
What is the probability it begins with ‘b’.
Probability
• Sample space, outcome, event, probability function. Rule: 𝑃 (𝐴 ∪𝐵) = 𝑃 (𝐴) +𝑃 (𝐵 )−
𝑃 (𝐴 ∩ 𝐵).
Special case: 𝑃 (𝐴𝑐) = 1 − 𝑃 (𝐴)
(𝐴 and 𝐵 disjoint ⇒ 𝑃 (𝐴 ∪ 𝐵) = 𝑃 (𝐴) + 𝑃 (𝐵).)
• Conditional probability, multiplication rule, trees, law of total probability, indepen-
dence
• Bayes’ theorem, base rate fallacy
Problem 4. Let 𝐸 and 𝐹 be two events. Suppose the probability that at least one of
them occurs is 2/3. What is the probability that neither 𝐸 nor 𝐹 occurs?
Problem 5. Let 𝐶 and 𝐷 be two events with 𝑃(𝐶 ) = 0.3, 𝑃 (𝐷) = 0.4, and 𝑃 (𝐶 𝑐∩ 𝐷) =
0.2.
What is 𝑃 (𝐶 ∩ 𝐷)?
Problem 6. Suppose we have 8 teams labeled 𝑇1, …, 𝑇8. Suppose they are ordered by
placing their names in a hat and drawing the names out one at a time.
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