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Probability Basics. Combinatorics. Bayes Theorem. Part 1 - nerjik.com

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Probability Basics. Combinatorics. Bayes Theorem. Part 1

Probability Basics. Combinatorics. Bayes Theorem. Part 1

Almaz Khusnutdinov

Almaz Khusnutdinov•Math enthusiast, love coding and mathematics

Published 9/27/2025

Conditional Probability

Problem 1.1

\text{In a family with 2 children, at least one is a girl. What is the probability that both are girls?}\\[8pt] \textbf{Answer: }\frac{1}{3}.

Problem 1.2

\text{A family has 2 children. One child chosen uniformly at random from the two is a boy. What is the probability that both children are boys?}\\[8pt] \textbf{Answer: }\frac{1}{2}.

Problem 1.3

\text{In a family with 3 children, at least one child is a boy. What is the probability that all three children are boys?}\\[8pt] \textbf{Answer: }\frac{1}{7}.

Problem 1.4

\text{Two independent servers A and B are up with probabilities }p_A=0.9,\ p_B=0.8.\ \text{The system is up iff at least one server is up.}\\ \text{Given the system is up, what is the probability that both servers are up?}\\[8pt] \textbf{Answer: }\frac{P(A\cap B)}{P(A\cup B)}=\frac{0.72}{0.9+0.8-0.72}=\frac{36}{49}.

Problem 1.5

\text{A die is selected: fair with probability } \tfrac{2}{3},\ \text{loaded with probability } \tfrac{1}{3}.\\ \text{Loaded die rolls }6\text{ with probability } \tfrac{1}{2}\ \text{and each of }1\text{--}5\text{ with probability } \tfrac{1}{10}.\\ \text{A single roll shows }6.\ \text{What is the probability the selected die was loaded?}\\[8pt] \textbf{Answer: }\frac{\tfrac{1}{3}\cdot\tfrac{1}{2}}{\tfrac{1}{3}\cdot\tfrac{1}{2}+\tfrac{2}{3}\cdot\tfrac{1}{6}}=\frac{3}{5}.

Problem 1.6

\text{Factory }F_1\text{ makes }40\%\text{ of chips with defect rate }1\%.\ \text{Factory }F_2\text{ makes }60\%\text{ with defect rate }3\%.\\ \text{A randomly chosen chip is defective. What is the probability it came from }F_1\text{?}\\[8pt] \textbf{Answer: }\frac{0.4\cdot0.01}{0.4\cdot0.01+0.6\cdot0.03}=\frac{2}{11}.

Problem 1.7

\text{A family has 2 children. It is known that at least one child is a boy born on a Tuesday. What is the probability that both children are boys?}\\[8pt] \textbf{Answer: }\frac{13}{27}.

Problem 1.8

\text{A family has 2 children. The older child is a boy. What is the probability that both children are boys?}\\[8pt] \textbf{Answer: }\frac{1}{2}.

Problem 1.9

\text{In a two-child family, each birth is a boy with probability }p\in(0,1)\text{ independently. Given that at least one child is a boy, what is the probability that both are boys?}\\[8pt] \textbf{Answer: }\displaystyle \frac{p^2}{1-(1-p)^2}=\frac{p}{2-p}.

Basic Probability

Problem 2.1

\text{A fair six-sided die is rolled twice.}\\[4pt] \text{(a) What is the probability that the sum of the two rolls is }9\text{?}\\ \text{(b) What is the probability that at least one of the rolls is }6\text{?}\\[10pt] \textbf{Solution to (a):}\\ \text{Each die has }6\text{ outcomes, so the sample space has }6\cdot 6=36\text{ equally likely outcomes.}\\[4pt] \text{We list all pairs with sum }9\text{: }(3,6),(4,5),(5,4),(6,3)\text{.}\\ \text{So there are }4\text{ favorable outcomes.}\\[4pt] P(\text{sum}=9)=\dfrac{4}{36}=\dfrac{1}{9}.\\[10pt] \textbf{Solution to (b):}\\ \text{Let }A=\{\text{at least one }6\}\text{. The complement }A^c\text{ is ``no 6 appears''.}\\ \text{On a single die, }P(\text{no }6)=\dfrac{5}{6}\text{. For two dice (independent):}\\ P(A^c)=P(\text{no }6\text{ on die 1 and no }6\text{ on die 2})=\dfrac{5}{6}\cdot\dfrac{5}{6}=\dfrac{25}{36}.\\[4pt] P(A)=1-P(A^c)=1-\dfrac{25}{36}=\dfrac{11}{36}.\\[10pt] \textbf{Answer: }(a)\ \dfrac{1}{9},\quad (b)\ \dfrac{11}{36}.

Problem 2.2

\text{A fair coin is tossed three times.}\\[4pt] \text{(a) What is the probability of getting exactly two heads?}\\ \text{(b) What is the probability of getting at least one tail?}\\[10pt] \textbf{Answer: }(a)\ \dfrac{3}{8},\quad (b)\ \dfrac{7}{8}.

Problem 2.3

\text{An urn contains }4\text{ red, }3\text{ blue, and }3\text{ green balls (}10\text{ in total). Two balls}\\ \text{are drawn at random without replacement.}\\[4pt] \text{(a) What is the probability that the two balls are of the same color?}\\ \text{(b) What is the probability that one ball is red and the other is blue?}\\[10pt] \textbf{Answer: }(a)\ \dfrac{4}{15},\quad (b)\ \dfrac{4}{15}.

Problem 2.4

\text{A standard deck of }52\text{ cards is well shuffled.}\\[4pt] \text{(a) One card is drawn. What is the probability that it is a heart or a king?}\\ \text{(b) Two cards are drawn without replacement. What is the probability that both}\\ \text{cards are aces?}\\[10pt] \textbf{Answer: }(a)\ \dfrac{4}{13},\quad (b)\ \dfrac{1}{221}.

Problem 2.5

\text{A class has }6\text{ boys and }4\text{ girls. Two students are chosen at random}\\ \text{without replacement.}\\[4pt] \text{(a) What is the probability that both chosen students are girls?}\\ \text{(b) What is the probability that at least one of the chosen students is a boy?}\\[10pt] \textbf{Answer: }(a)\ \dfrac{2}{15},\quad (b)\ \dfrac{13}{15}.

Problem 2.6

\text{Two fair six-sided dice are rolled.}\\[4pt] \text{(a) Given that the sum of the two dice is }7\text{, what is the probability that}\\ \text{one of the dice shows }3\text{?}\\ \text{(b) Given that at least one die shows }5\text{, what is the probability that}\\ \text{the sum of the two dice is }10\text{?}\\[10pt] \textbf{Answer: }(a)\ \dfrac{1}{3},\quad (b)\ \dfrac{1}{11}.

Problem 2.7

\text{A bag contains }5\text{ white balls and }7\text{ black balls (}12\text{ in total). Three balls}\\ \text{are drawn at random without replacement.}\\[4pt] \text{(a) What is the probability that exactly one of the three balls is white?}\\ \text{(b) What is the probability that at least two of the three balls are white?}\\[10pt] \textbf{Answer: }(a)\ \dfrac{21}{44},\quad (b)\ \dfrac{4}{11}.

Problem 2.8

\text{Two digits are chosen independently and uniformly from }\{0,1,2,\dots,9\}\text{.}\\ \text{Let the ordered pair be }(D_1,D_2)\text{.}\\[4pt] \text{(a) What is the probability that }D_1+D_2=10\text{?}\\ \text{(b) What is the probability that }D_1>D_2\text{?}\\[10pt] \textbf{Answer: }(a)\ \dfrac{9}{100},\quad (b)\ \dfrac{9}{20}.

Problem 2.9

\text{A box contains }8\text{ good light bulbs and }2\text{ defective ones. Three bulbs}\\ \text{are chosen at random without replacement.}\\[4pt] \text{(a) What is the probability that none of the chosen bulbs is defective?}\\ \text{(b) What is the probability that at least one of the chosen bulbs is defective?}\\[10pt] \textbf{Answer: }(a)\ \dfrac{7}{15},\quad (b)\ \dfrac{8}{15}.

Problem 2.10

\text{A single card is drawn at random from a standard }52\text{-card deck.}\\[4pt] \text{(a) Given that the card is red, what is the probability that it is a heart?}\\ \text{(b) Given that the card is a face card (J, Q, or K), what is the probability}\\ \text{that it is a king?}\\[10pt] \textbf{Answer: }(a)\ \dfrac{1}{2},\quad (b)\ \dfrac{1}{3}.

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