Finding Conditional Probability in Models - Statistics and Probability

1. Find $P(A\mid B)$ if $|A\cap B|=18$ and $|B|=60$ in an equally likely model.

   $0.30$. Divide favorable count by total in $B$: $\frac{18}{60}=0.30$.

2. Find and correct the error: $P(A\mid B)=\frac{P(A)}{P(B)}$.

   Correct: $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$. Must use intersection, not just $P(A)$.

3. What is $P(A\mid B)$ if $B\subseteq A$ and $P(B)>0$?

   $1$. All of $B$ is in $A$, so $P(A\cap B)=P(B)$.

4. In a two-way table, which cell count is the numerator for $P(A\mid B)$?

   The intersection count $\#(A\cap B)$. Counts outcomes satisfying both conditions.

5. State the formula for $P(A\mid B)$ using probabilities (not outcome counts).

   $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$, with $P(B)>0$. Divides joint probability by the condition's probability.

6. If $A\subseteq B$ and $P(B)>0$, what is $P(A\mid B)$ in terms of $P(A)$ and $P(B)$?

   $\frac{P(A)}{P(B)}$. When $A\subseteq B$, $P(A\cap B)=P(A)$.

7. What is the relationship between $P(A\cap B)$ and $P(A\mid B)$?

   $P(A\cap B)=P(A\mid B)\,P(B)$. Multiplication rule for joint probabilities.

8. A bag has $10$ red and $6$ blue marbles; $4$ red are large and $3$ blue are large. Find $P(\text{red}\mid \text{large})$.

   $\frac{4}{7}$. Among $7$ large marbles, $4$ are red.

9. A class has $20$ students; $8$ are in band, and $5$ are in band and play a sport. Find $P(\text{sport}\mid \text{band})$.

   $\frac{5}{8}$. Among $8$ band students, $5$ play sports.

10. Find $P(A\cap B)$ if $P(A\mid B)=0.20$ and $P(B)=0.50$.

    $0.10$. Multiply: $0.20 \times 0.50 = 0.10$.

11. Identify the condition required for $P(A\mid B)$ to be defined.

    $P(B)>0$. Cannot divide by zero probability.

12. What does $P(A\mid B)$ mean in words in terms of outcomes in $B$?

    Probability of $A$ among outcomes restricted to $B$. Restricts sample space to only outcomes in $B$.

13. State the fraction interpretation of $P(A\mid B)$ using counts $|A\cap B|$ and $|B|$.

    $P(A\mid B)=\frac{|A\cap B|}{|B|}$ (equally likely outcomes). Counts favorable outcomes within the restricted set $B$.

14. State the multiplication rule that rewrites $P(A\cap B)$ using $P(A\mid B)$.

    $P(A\cap B)=P(A\mid B)P(B)$. Rearranges conditional probability formula.

15. Which expression equals $P(A\mid B)$: $\frac{P(A\cap B)}{P(B)}$ or $\frac{P(A\cup B)}{P(B)}$?

    $\frac{P(A\cap B)}{P(B)}$. Uses intersection, not union, in numerator.

16. If events $A$ and $B$ are disjoint and $P(B)>0$, what is $P(A\mid B)$?

    $0$. Disjoint events have empty intersection.

17. Identify the correct equality for independent events using conditional probability.

    If independent, then $P(A\mid B)=P(A)$. Independence means conditioning doesn't change probability.

18. Find $P(B)$ if $P(A\cap B)=0.18$ and $P(A\mid B)=0.60$.

    $0.30$. Rearrange: $P(B)=\frac{0.18}{0.60}=0.30$.

19. State the formula for conditional probability $P(A\mid B)$ using intersection and $P(B)$.

    $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$ for $P(B)>0$. Divides joint probability by the condition's probability.

20. Choose the correct interpretation: If $P(A\mid B)=0.7$, what does $0.7$ represent?

    Among outcomes in $B$, the fraction that also satisfy $A$ is $0.7$. Proportion of $B$ outcomes that also satisfy $A$.

21. What does $P(A\mid B)$ mean in words in a probability model?

    Probability that $A$ occurs given that $B$ has occurred. Updates probability based on knowing $B$ happened.

22. What is the definition of conditional probability $P(A\mid B)$ in terms of outcomes?

    $P(A\mid B)=\frac{\#(A\cap B)}{\#(B)}$ for equally likely outcomes. Counts favorable outcomes in $B$ that also satisfy $A$.

23. A jar has $10$ marbles: $6$ red, $4$ blue; $3$ red are striped, $1$ blue is striped. Find $P(\text{striped}\mid \text{red})$.

    $\frac{3}{6}=\frac{1}{2}$. Among $6$ red marbles, $3$ are striped.

24. Compute $P(A\mid B)$ from a table where $|A\cap B|=25$ and $|B|=100$.

    $0.25$. Fraction of $B$'s outcomes in $A$: $\frac{25}{100}=0.25$.

25. A table shows $\#(A\cap B)=9$ and $\#(B)=15$. What is $P(A\mid B)$?

    $\frac{9}{15}=\frac{3}{5}$. Direct calculation using table counts.

26. Compute $P(A\mid B)$ if $A$ and $B$ are independent and $P(A)=0.35$.

    $0.35$. Independence implies $P(A\mid B)=P(A)$.

27. If $B\subseteq A$ and $P(B)>0$, what is $P(A\mid B)$?

    $1$. If $B\subseteq A$, all $B$ outcomes are in $A$.

28. State the multiplication rule that rewrites $P(A\cap B)$ using $P(B\mid A)$.

    $P(A\cap B)=P(B\mid A)P(A)$. Same rule with roles of $A$ and $B$ reversed.

29. In a class, $12$ students are left-handed; $5$ are left-handed and wear glasses. Find $P(\text{glasses}\mid \text{left})$.

    $\frac{5}{12}$. Among $12$ left-handed, $5$ wear glasses.

30. What condition must hold for $P(A\mid B)$ to be defined?

    $P(B)>0$. Cannot divide by zero probability.

31. In a two-way table, which total count is the denominator for $P(A\mid B)$?

    The $B$ row (or $B$ column) total $\#(B)$. Total count of the given condition.

32. Find $P(A\mid B)$ if $\#(A\cap B)=12$ and $\#(B)=30$ (equally likely outcomes).

    $\frac{12}{30}=\frac{2}{5}$. Direct application of outcome-based formula.

33. Identify the correct interpretation: If $P(A\mid B)=0.70$, what does $0.70$ describe?

    $70\%$ of outcomes in $B$ are also in $A$. Fraction of $B$'s outcomes that belong to $A$.

34. What is $P(B\mid A)$ in terms of $P(A\cap B)$ and $P(A)$?

    $P(B\mid A)=\frac{P(A\cap B)}{P(A)}$, with $P(A)>0$. Swaps roles of $A$ and $B$ in conditional formula.

35. What is $P(A\mid B)$ if $A$ and $B$ are disjoint and $P(B)>0$?

    $0$. Disjoint means $A\cap B=\emptyset$, so $P(A\cap B)=0$.

36. Find $P(A\cap B)$ if $P(A\mid B)=0.25$ and $P(B)=0.40$.

    $0.25\times 0.40=0.10$. Uses multiplication rule $P(A\cap B)=P(A\mid B)P(B)$.

37. Choose the correct denominator for $P(A\mid B)$ in a two-way table: $|A\cap B|$, $|A|$, or $|B|$?

    Denominator is $|B|$. Restricts to outcomes in condition $B$.

38. Identify the sample space used when computing $P(A\mid B)$ from equally likely outcomes.

    The restricted sample space consisting only of outcomes in $B$. Conditioning restricts to outcomes where $B$ occurs.

Study these flashcards