Statistics Flashcards: Analyzing Decisions And Strategies With Probability

What this deck covers

This deck focuses on Analyzing Decisions And Strategies With Probability, giving you a quick way to review the definitions, rules, and examples that matter most for Statistics.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is sensitivity in terms of conditional probability?

Answer: $\text{sensitivity}=P(+\mid D)$. Sensitivity is probability of positive test given disease.

Flashcard 2: What is the specificity of a test in probability notation?

Answer: $P(\text{test}-\mid\text{no disease})$. Probability of negative test given no disease.

Flashcard 3: What is the complement rule for an event $A$?

Answer: $P(A^c)=1-P(A)$. Complement probability is one minus the event's probability.

Flashcard 4: What is $P(D\mid +)$ if $P(D)=0.50$, $P(+\mid D)=0.90$, and $P(+\mid D^c)=0.10$?

Answer: $0.90$. Apply Bayes' theorem: $\frac{0.5(0.9)}{0.5(0.9)+0.5(0.1)}=\frac{0.45}{0.5}$.

Flashcard 5: What is the definition of independence using conditional probability?

Answer: $A,B$ independent iff $P(A\mid B)=P(A)$. Independence means conditioning doesn't change the probability.

Flashcard 6: What is the definition of a false negative in a medical test?

Answer: Test indicates no disease when the person actually has the disease. Negative result despite disease being present.

Flashcard 7: What is the negative predictive value (NPV) in probability notation?

Answer: $\text{NPV}=P(D^c\mid -)$. NPV is probability of no disease given negative test.

Flashcard 8: What is $E(X)$ if you gain $100 with prob $0.02$ and lose $5 with prob $0.98$?

Answer: -\3$.$E=0.02(100)+0.98(-5)=2-4.9=-2.9\approx-3$.

Flashcard 9: What is the sensitivity of a test in probability notation?

Answer: $P(\text{test}+\mid\text{disease})$. Probability of positive test given disease is present.

Flashcard 10: What is the conditional probability formula for $P(A\mid B)$?

Answer: $P(A\mid B)=\frac{P(A \cap B)}{P(B)}$. Conditional probability is joint probability divided by condition's probability.

Flashcard 11: What is the definition of a false positive in a medical test?

Answer: Test indicates disease when the person is actually disease-free. Positive result despite no disease present.

Flashcard 12: What is the positive predictive value (PPV) in probability notation?

Answer: $\text{PPV}=P(D\mid +)$. PPV is probability of disease given positive test.

Flashcard 13: A goalie-pull choice: If pulled, win prob is $0.12$; if not, win prob is $0.06$. Which strategy maximizes win probability?

Answer: Pull the goalie. Pulling doubles win probability from 6% to 12%.

Flashcard 14: What is $P(D^c\mid -)$ if $P(D)=0.20$, $P(-\mid D)=0.10$, and $P(-\mid D^c)=0.90$?

Answer: $\frac{36}{41}$. Apply Bayes' theorem: $\frac{0.8(0.9)}{0.2(0.1)+0.8(0.9)}=\frac{0.72}{0.82}$.

Flashcard 15: What is $P(D\mid +)$ if $P(D)=0.10$, $P(+\mid D)=0.90$, and $P(+\mid D^c)=0.20$?

Answer: $\frac{1}{3}$. Apply Bayes' theorem: $\frac{0.1(0.9)}{0.1(0.9)+0.9(0.2)}=\frac{0.09}{0.27}$.

Flashcard 16: Which option is larger if $P(D)=0.01$, $P(+\mid D)=0.99$, $P(+\mid D^c)=0.05$: $P(D\mid +)$ or $0.5$?

Answer: $P(D\mid +)<0.5$. Low disease prevalence makes PPV small despite high sensitivity.

Flashcard 17: What is the expected value formula for a discrete outcome variable $X$?

Answer: $E(X)=\sum x\,P(X=x)$. Expected value sums products of outcomes and their probabilities.

Flashcard 18: What is the definition of a false negative in a medical test?

Answer: Test negative when the person actually has the disease. False negative occurs when diseased person tests negative.

Flashcard 19: Identify the posterior odds form: what is $\frac{P(A\mid B)}{P(A^c\mid B)}$ equal to?

Answer: $\frac{P(B\mid A)}{P(B\mid A^c)}\cdot\frac{P(A)}{P(A^c)}$. Likelihood ratio times prior odds equals posterior odds.

Flashcard 20: Which option maximizes expected value: A pays $10 with prob $0.4$; B pays $6 with prob $0.7$?

Answer: Option B. Option A: $E=0.4(10)=4$; Option B: $E=0.7(6)=4.2$.

Flashcard 21: What is the positive predictive value (PPV) in probability notation?

Answer: $P(\text{disease}\mid\text{test}+)$. Probability of having disease given positive test.

Flashcard 22: Which strategy has higher expected goals: pull goalie gives $0.12$ for, $0.30$ against; keep gives $0.05$ for, $0.10$ against?

Answer: Keep the goalie in. Pull: $E=0.12-0.30=-0.18$; Keep: $E=0.05-0.10=-0.05$.

Flashcard 23: What is the negative predictive value (NPV) in probability notation?

Answer: $P(\text{no disease}\mid\text{test}-)$. Probability of being healthy given negative test.

Flashcard 24: Compute expected value: Option A pays $100 with prob $0.3$ else $0; Option B pays $20 for sure. Which has higher $E$?

Answer: Option A, since $E_A=100\cdot^0.3=30>20=E_B$. Expected value calculation favors risky option over sure $20.

Flashcard 25: What is the expected value formula for a discrete outcome with values $x_i$ and probabilities $p_i$?

Answer: $E(X)=\sum x_i p_i$. Weighted average of outcomes by their probabilities.

Flashcard 26: State Bayes' rule for $P(A\mid B)$ using $P(B\mid A)$, $P(A)$, and $P(B)$.

Answer: $P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}$. Updates prior probability using new evidence.

Flashcard 27: What is the definition of a false positive in a medical test?

Answer: Test positive when the person is actually disease-free. False positive occurs when healthy person tests positive.

Flashcard 28: What is the addition rule for any events $A$ and $B$?

Answer: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. Union probability adds individual probabilities, subtracts overlap.

Flashcard 29: What is the multiplication rule for independent events $A$ and $B$?

Answer: $P(A \cap B)=P(A)P(B)$. For independent events, joint probability equals product of individual probabilities.

Flashcard 30: What is the general multiplication rule for events $A$ and $B$?

Answer: $P(A \cap B)=P(A)P(B\mid A)$. Joint probability equals first event's probability times conditional probability.

Flashcard 31: Which decision rule chooses the action with the larger expected value (or smaller expected cost)?

Answer: Choose the option with the higher expected value. Maximizes expected gain or minimizes expected loss.

Flashcard 32: What is the multiplication rule for joint probability $P(A\cap B)$?

Answer: $P(A\cap B)=P(A)P(B\mid A)$. Joint probability equals marginal times conditional.

Flashcard 33: What is specificity in terms of conditional probability?

Answer: $\text{specificity}=P(-\mid D^c)$. Specificity is probability of negative test given no disease.

Flashcard 34: What is the complement rule for an event $A$?

Answer: $P(A^c)=1-P(A)$. Complement probability is one minus the event probability.