Understanding Independent Events - Statistics and Probability

1. Identify whether $A$ and $B$ are independent if $P(A)=\frac{2}{5}$, $P(B)=\frac{1}{2}$, and $P(A\cap B)=\frac{1}{5}$.

   Independent. Check: $\frac{2}{5} \times \frac{1}{2} = \frac{1}{5}$, which equals $P(A \cap B)$.

2. Which statement is always true about independence: does $A$ being independent of $B$ imply $B$ independent of $A$?

   Yes, independence is symmetric. If $P(A\cap B)=P(A)P(B)$, then both directions hold.

3. Identify whether $A$ and $B$ are independent if $P(A)=\frac{1}{3}$, $P(B)=\frac{1}{2}$, and $P(A\cap B)=\frac{1}{6}$.

   Independent. $\frac{1}{3} \times \frac{1}{2} = \frac{1}{6}$, which equals $P(A\cap B)$.

4. What equation tests independence using $P(A\mid B)$ when $P(B)>0$?

   $P(A\mid B)=P(A)$. If $B$ doesn't affect $A$'s probability, they're independent.

5. What is the independence condition using the product rule for events $A$ and $B$?

   $P(A \cap B)=P(A)P(B)$. Independent events satisfy this multiplication rule.

6. Find $P(B)$ if $A$ and $B$ are independent with $P(A)=0.25$ and $P(A\cap B)=0.05$.

   $0.2$. Solve: $0.25 \times P(B) = 0.05$, so $P(B) = 0.2$.

7. What is the independence test written using conditional probability $P(B\mid A)$?

   $P(B\mid A)=P(B)$. For independent events, knowing A occurred doesn't change the probability of B.

8. Identify whether $A$ and $B$ are independent from this table: $P(A)=0.4$, $P(B)=0.3$, $P(A\cap B)=0.10$.

   Not independent. $0.4 \times 0.3 = 0.12 \neq 0.10$.

9. Find and correct the mistake: claiming independence because $P(A\cap B)=P(A)+P(B)$.

   Correct test: $P(A\cap B)=P(A)P(B)$. Addition rule is for disjoint events, not independence.

10. Find $P(B)$ if $A$ and $B$ are independent, $P(A)=0.25$, and $P(A\cap B)=0.05$.

    $0.20$. Solve $0.25 \times P(B) = 0.05$ to get $P(B) = 0.20$.

11. Identify whether $A$ and $B$ are independent if $P(A)=0.4$, $P(B)=0.5$, and $P(A\cap B)=0.2$.

    Independent. $0.4 \times 0.5 = 0.2$, which equals $P(A\cap B)$.

12. Identify whether $A$ and $B$ are independent if $P(A)=\frac{2}{5}$, $P(B)=\frac{1}{2}$, and $P(A\cap B)=\frac{1}{4}$.

    Not independent. Check: $\frac{2}{5} \times \frac{1}{2} = \frac{1}{5} \neq \frac{1}{4}$.

13. Identify whether $A$ and $B$ are independent if $P(A)=0.3$, $P(B)=0.6$, and $P(A\cap B)=0.25$.

    Not independent. $0.3 \times 0.6 = 0.18 \neq 0.25$.

14. What is the defining equation for independence using the intersection probability?

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

15. Find $P(A\cap B)$ if $A$ and $B$ are independent with $P(A)=\frac{3}{4}$ and $P(B)=\frac{2}{3}$.

    $\frac{1}{2}$. Multiply: $\frac{3}{4} \times \frac{2}{3} = \frac{6}{12} = \frac{1}{2}$.

16. Find $P(A)$ if $A$ and $B$ are independent, $P(B)=0.8$, and $P(A\cap B)=0.12$.

    $0.15$. Solve $P(A) \times 0.8 = 0.12$ to get $P(A) = 0.15$.

17. Find $P(B\mid A)$ if $A$ and $B$ are independent and $P(B)=0.12$.

    $0.12$. For independent events, $P(B \mid A) = P(B)$.

18. What is the symmetric conditional form of independence between events $A$ and $B$?

    $P(A\mid B)=P(A)$ and $P(B\mid A)=P(B)$. Independence means each event's probability is unchanged by the other.

19. Identify whether $A$ and $B$ are independent if $P(A)=0.6$, $P(B)=0.3$, and $P(A\cap B)=0.25$.

    Not independent. Check: $0.6 \times 0.3 = 0.18 \neq 0.25$.

20. Find $P(A\mid B)$ if $A$ and $B$ are independent and $P(A)=0.35$.

    $0.35$. For independent events, $P(A \mid B) = P(A)$.

21. Identify whether $A$ and $B$ are independent if $P(A)=0.5$, $P(B)=0.4$, and $P(A\mid B)=0.5$.

    Independent. Since $P(A \mid B) = P(A) = 0.5$, they are independent.

22. Which formula gives conditional probability $P(B\mid A)$ in terms of $P(A\cap B)$ and $P(A)$?

    $P(B\mid A)=\frac{P(A\cap B)}{P(A)}$. Conditional probability is the ratio of joint to marginal probability.

23. Find $P(A\mid B)$ if $A$ and $B$ are independent and $P(A)=0.65$.

    $0.65$. For independent events, $P(A\mid B) = P(A)$.

24. Which statement correctly describes independence: $P(A\cap B)>P(A)P(B)$, $=$, or $<$?

    $P(A\cap B)=P(A)P(B)$. Independence requires equality, not greater than or less than.

25. Identify whether $A$ and $B$ are independent if $P(A)=0.5$ and $P(A\mid B)=0.5$ with $P(B)>0$.

    Independent. $P(A\mid B) = P(A)$ confirms independence.

26. Which statement is true if $A$ and $B$ are independent: $P(A\cap B)$ equals what expression?

    $P(A\cap B)=P(A)P(B)$. This is the defining property of independent events.

27. What is $P(A \cap B)$ if $A$ and $B$ are independent and you know $P(A)$ and $P(B)$?

    $P(A \cap B)=P(A)P(B)$. For independent events, multiply their individual probabilities.

28. Find $P(A)$ if $A$ and $B$ are independent with $P(B)=0.8$ and $P(A\cap B)=0.24$.

    $0.3$. Solve: $P(A) \times 0.8 = 0.24$, so $P(A) = 0.3$.

29. Which probability must equal $P(A)P(B)$ for events $A$ and $B$ to be independent?

    $P(A \cap B)$. The intersection probability must equal the product for independence.

30. What equation tests independence using $P(B\mid A)$ when $P(A)>0$?

    $P(B\mid A)=P(B)$. If $A$ doesn't affect $B$'s probability, they're independent.

31. Identify whether $A$ and $B$ are independent if $P(A)=0.5$, $P(B)=0.4$, and $P(A\mid B)=0.6$.

    Not independent. Since $P(A \mid B) = 0.6 \neq 0.5 = P(A)$, they're not independent.

32. What is the equivalent independence condition using $P(A\cap B)$ and $P(B)$ when $P(B)>0$?

    $P(A\mid B)=\frac{P(A\cap B)}{P(B)}=P(A)$. Rearranging the conditional probability formula shows independence.

33. Find $P(A\cap B)$ if $A$ and $B$ are independent with $P(A)=0.7$ and $P(B)=0.2$.

    $0.14$. Multiply: $0.7 \times 0.2 = 0.14$.

34. What is the independence test written using conditional probability $P(A\mid B)$?

    $P(A\mid B)=P(A)$. For independent events, knowing B occurred doesn't change the probability of A.

35. Find $P(A\cap B)$ if $A$ and $B$ are independent, $P(A)=0.7$, and $P(B)=0.2$.

    $0.14$. For independent events, multiply: $0.7 \times 0.2 = 0.14$.

36. Identify whether $A$ and $B$ are independent if $P(A)=0.5$ and $P(A\mid B)=0.7$ with $P(B)>0$.

    Not independent. $P(A\mid B) \neq P(A)$ means not independent.

37. Which formula gives conditional probability $P(A\mid B)$ in terms of $P(A\cap B)$ and $P(B)$?

    $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$. Conditional probability is the ratio of joint to marginal probability.

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