Conditional Probability and Independence: Determine Independence ...
Conditional Probability and Independence: Determine Independence of Events using Conditional Probabilities Checks for Understanding 1. Call a household prosperous if its income exceeds $100,000. Call the household educated if the householder completed college. Select an American household at random, and let A be the event that the selected household is prosperous and B the event that it is educated. According to the Current Population Survey, P(A) = 0.138, P(B) = 0.261, and the probability that a household is both prosperous and educated is P(A∩B) = 0.082. a. What is the conditional probability that a household is prosperous, given that is educated? b. Explain why your result shows that events A and B are not independent.
2. Here are the counts (in thousands) of earned degrees in the United States in the 2005-2006 academic year, classified by level and by the sex of the degree recipient:
Are the events “choose a woman” and “choose a professional degree recipient” independent? How do you know?
3. If Event A and Event B are independent, which of the following are true? (Select all that apply). a. P(A) > P(A|B) b. P(A) = P(A|B) c. P(A) < P(A|B) d. Events A and B are mutually exclusive e. P(A|B) = P (B|A)
Answers 1a.
P(A|B) [the probability of event A occurring given that event B occurred] is equal to
𝑃(𝐴∩𝐵) 𝑃(𝐵)
.
0.082
In this case, 0.261 = 0.3142. So, the conditional probability that a household is prosperous, given that it is educated, is .3142, or approximately 31.42%.
1b. If A and B were independent, P(A|B) would equal P(A). Why? Because the probability of event A occurring would not change at all given that B occurs. In other words, event B has no effect on the chances of event A occurring. Also, P(A∩B) would equal P(A) ∗ P(B).
2. “Choose a woman” [let’s call this P(W)] and “Choose a professional degree recipient” [let’s call this P(W|P) ARE NOT independent. P(W) = 1119/1944 = 0.5756 P(W|P) = 39/83 = 0.4699 If these two events were independent, then the two probabilities that I calculated would have to be equal, because the chances of W occurring should not be affected by P occurring.
3. a. False b. True! Remember, if two events are independent, then the likelihood of a single event occurring stays the same, even if we know that the other event occurred. This also explains why part a and part b are false. c. False d. False. Mutually exclusive means that the two events cannot both occur. With mutually exclusive events, if one event happens, then the chances of the other event occurring change to zero (or impossible). This violates the law of independent events that the probability stays the same whether or not the other event occurred. Therefore, independent events are NEVER mutually exclusive. False. This can be tricky. Think about what this means. It really means that event A and event B are equally likely. Independent events do not have to be like a coin-flip where there is a 50/50 chance of the two events occurring. They can have different probabilities, and so that is why this statement is false.
2. Here are the counts (in thousands) of earned degrees in the United States in the 2005-2006 academic year, classified by level and by the sex of the degree recipient:
Are the events “choose a woman” and “choose a professional degree recipient” independent? How do you know?
3. If Event A and Event B are independent, which of the following are true? (Select all that apply). a. P(A) > P(A|B) b. P(A) = P(A|B) c. P(A) < P(A|B) d. Events A and B are mutually exclusive e. P(A|B) = P (B|A)
Answers 1a.
P(A|B) [the probability of event A occurring given that event B occurred] is equal to
𝑃(𝐴∩𝐵) 𝑃(𝐵)
.
0.082
In this case, 0.261 = 0.3142. So, the conditional probability that a household is prosperous, given that it is educated, is .3142, or approximately 31.42%.
1b. If A and B were independent, P(A|B) would equal P(A). Why? Because the probability of event A occurring would not change at all given that B occurs. In other words, event B has no effect on the chances of event A occurring. Also, P(A∩B) would equal P(A) ∗ P(B).
2. “Choose a woman” [let’s call this P(W)] and “Choose a professional degree recipient” [let’s call this P(W|P) ARE NOT independent. P(W) = 1119/1944 = 0.5756 P(W|P) = 39/83 = 0.4699 If these two events were independent, then the two probabilities that I calculated would have to be equal, because the chances of W occurring should not be affected by P occurring.
3. a. False b. True! Remember, if two events are independent, then the likelihood of a single event occurring stays the same, even if we know that the other event occurred. This also explains why part a and part b are false. c. False d. False. Mutually exclusive means that the two events cannot both occur. With mutually exclusive events, if one event happens, then the chances of the other event occurring change to zero (or impossible). This violates the law of independent events that the probability stays the same whether or not the other event occurred. Therefore, independent events are NEVER mutually exclusive. False. This can be tricky. Think about what this means. It really means that event A and event B are equally likely. Independent events do not have to be like a coin-flip where there is a 50/50 chance of the two events occurring. They can have different probabilities, and so that is why this statement is false.
