For each set of probabilities, determine whether the events A and B are independent or dependent.

Answers:

Explanation:

If events A and B are independent, then the two following equations must both be true

This is because the conditional probability P(A|B) means "P(A) when B has happened". If B were to happen, then P(A) must be the same as before. In other words, event B does not affect A, and vice versa.

For part (a), we have P(B) = 1/4 and P(B|A) = 1/4 showing that P(B|A) = P(B) is true, and therefore we can say the events are independent. We don't need the info that P(A) = 1/8.

Unlike part (a), part (b) has the answer "dependent" because P(A) = 1/8 and P(A | B) = 1/3 differ in value. Event A starts off at probability 1/8, but then event B occurring means P(A) gets increased to 1/3. The prior knowledge about B changes the chances of A. The P(B) = 1/5 is unneeded.

If A and B were independent, then,

P(A and B) = P(A)*P(B)

However,

P(A)*P(B) = (1/4)*(1/5) = 1/20

which is not the same as P(A and B) = 1/6. Therefore the two events are dependent.

Refer back to part (a)

P(A) = 1/4 and P(A|B) = 1/4 are identical in value, so P(A|B) = P(A) which leads to the events being independent. Whether we know event B happened or not, it does not affect the outcome of event A. P(B) = 1/9 is unneeded.

7. d. 64

8. i. 32

9. c. 125

10. j. none of the above

11. d. 81

12. i. 216

13. e. none of the above

14. g. 240

15. e. none of the above

16. g. 32

17.   e. none of the above

answer: he is 272195

step-by-step explanation: all you have to do is multiple 3535 and 77 and it equals 272195