Probability - Quantitative Reasoning

1. Identify $P(A\cap B)$ if $P(A)=0.3$ and $P(B\mid A)=0.5$.

   $0.15$. Apply the multiplication rule: $P(A\cap B) = 0.3 \times 0.5 = 0.15$.

2. What is the definition of probability for an event $E$ in a finite equally likely sample space?

   $P(E)=\frac{\#(E)}{\#(S)}$. In a finite sample space where all outcomes are equally likely, the probability of event $E$ is the number of favorable outcomes divided by the total number of outcomes.

3. What is the formula for probability of "at least one" occurrence of event $A$ in repeated trials?

   $P(\ge 1)=1-P(0)$. The probability of at least one occurrence is one minus the probability of zero occurrences in independent trials.

4. Find $P(A^c)$ given $P(A)=0.13$.

   $0.87$. Use the complement rule: $P(A^c) = 1 - 0.13 = 0.87$.

5. What is the multiplication rule using conditional probability for events $A$ and $B$?

   $P(A\cap B)=P(A)P(B\mid A)$. The multiplication rule expresses the joint probability as the product of one event's probability and the conditional probability of the other given the first.

6. What is the counting formula for combinations of $n$ distinct items chosen $r$ at a time?

   $\binom{n}{r}=\frac{n!}{r!(n-r)!}$. Combinations count the number of ways to choose $r$ items out of $n$ distinct ones, where order does not matter.

7. What is the probability of drawing $2$ red cards in $2$ draws with replacement from a $52$-card deck?

   $\frac{1}{4}$. With replacement, $P($both red$) = \frac{26}{52} \times \frac{26}{52} = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$.

8. What is the counting formula for permutations of $n$ distinct items taken $r$ at a time?

   $P(n,r)=\frac{n!}{(n-r)!}$. Permutations count the number of ways to arrange $r$ items out of $n$ distinct ones, where order matters.

9. Identify $P(A\mid B)$ given $P(A\cap B)=0.12$ and $P(B)=0.3$.

   $0.4$. Use the conditional probability formula: $P(A\mid B) = \frac{0.12}{0.3} = 0.4$.

10. What is the addition rule for two events $A$ and $B$ (not necessarily disjoint)?

    $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. The addition rule accounts for the overlap by subtracting the intersection probability from the sum of individual probabilities.

11. Identify $P(A\cup B)$ given $P(A)=0.4$, $P(B)=0.5$, and $P(A\cap B)=0.2$.

    $0.7$. Apply the addition rule: $P(A\cup B) = 0.4 + 0.5 - 0.2 = 0.7$.

12. Identify $E[X]$ for $X\sim\text{Bin}(10,0.3)$.

    $3$. For binomial, $E[X] = np = 10 \times 0.3 = 3$.

13. What is the formula for the variance of $X\sim\text{Bin}(n,p)$?

    $\text{Var}(X)=np(1-p)$. The variance of a binomial random variable measures spread as $np$ times the failure probability.

14. What is the probability of at least one head in $3$ fair coin flips?

    $\frac{7}{8}$. Calculate as $1 - P($all tails$) = 1 - (\frac{1}{2})^3 = 1 - \frac{1}{8} = \frac{7}{8}$.

15. What is the probability of getting at least one $6$ in $2$ fair die rolls?

    $\frac{11}{36}$. Calculate as $1 - P($no 6$) = 1 - (\frac{5}{6})^2 = 1 - \frac{25}{36} = \frac{11}{36}$.

16. What is the formula for conditional probability $P(A\mid B)$?

    $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$. Conditional probability measures the likelihood of $A$ occurring given $B$ has occurred, by dividing the joint probability by $P(B)$.

17. Identify $P(A\cap B)$ if $A$ and $B$ are independent with $P(A)=0.2$ and $P(B)=0.6$.

    $0.12$. For independent events, $P(A\cap B) = 0.2 \times 0.6 = 0.12$.

18. What is the addition rule for mutually exclusive events $A$ and $B$?

    $P(A\cup B)=P(A)+P(B)$. For mutually exclusive events, there is no overlap, so their union probability is simply the sum of their individual probabilities.

19. What is the formula for the expected value of a binomial random variable $X\sim\text{Bin}(n,p)$?

    $E[X]=np$. The expected value of a binomial random variable is the product of the number of trials and the success probability per trial.

20. What condition defines independence of events $A$ and $B$ using an equation?

    $P(A\cap B)=P(A)P(B)$. Events $A$ and $B$ are independent if their joint probability equals the product of their marginal probabilities.

21. What is $P(A\mid B)$ if $A$ and $B$ are independent and $P(A)$ is known?

    $P(A\mid B)=P(A)$. For independent events, the occurrence of $B$ does not affect the probability of $A$, so the conditional equals the unconditional probability.

22. What is the formula for the probability of exactly $k$ successes in $n$ Bernoulli trials?

    $\binom{n}{k}p^k(1-p)^{n-k}$. The binomial probability formula gives the likelihood of exactly $k$ successes in $n$ independent trials each with success probability $p$.

23. What is the probability of exactly $2$ heads in $3$ fair coin flips?

    $\frac{3}{8}$. Use binomial formula: $\binom{3}{2} (\frac{1}{2})^3 = 3 \times \frac{1}{8} = \frac{3}{8}$.

24. What is the complement rule for an event $A$?

    $P(A^c)=1-P(A)$. The complement rule states that the probability of an event not occurring equals one minus the probability of it occurring.

Study these flashcards