Question

Suppose that \(P(A)=.3\) and \(P(B)=.4\) a. If \(P(A \cap B)=.12,\) are \(A\) and \(B\) independent? Justify your answer. b. If \(P(A \cup B)=.7,\) what is \(P(A \cap B)\) ? Justify your answer. c. If \(A\) and \(B\) are independent, what is \(P(A \mid B)\) ? d. If \(A\) and \(B\) are mutually exclusive, what is \(P(A \mid B) ?\)

Short answer

In summary, we determined that events A and B are independent, since their intersection probability equals the product of their individual probabilities. Given the probabilities provided, \(P(A \cap B) = 0.12\). If A and B are independent, the conditional probability \(P(A \mid B) = 0.3\), while if A and B are mutually exclusive, the conditional probability \(P(A \mid B) = 0\).

Step-by-step solution

Determine if A and B are independent

Two events A and B are independent if their intersection probability is equal to the product of their individual probabilities: \(P(A \cap B) = P(A)P(B)\). If this condition is not met, they are dependent events. Given \(P(A)=.3\), \(P(B)=.4\), and \(P(A \cap B)=.12\). Let's check if \(A\) and \(B\) are independent: \(P(A)P(B) = (0.3)(0.4) = 0.12\) The intersection probability equals the product of the individual probabilities, so A and B are independent events. a) Yes, events A and B are independent, because their intersection probability equals the product of their individual probabilities: \(P(A \cap B) = P(A)P(B)\).

Find \(P(A \cap B)\) using the formula for \(P(A \cup B)\)

To find \(P(A \cap B)\) given \(P(A \cup B)\) we can use the following formula: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\). Rearranging this formula, we can find \(P(A \cap B)\): \(P(A \cap B) = P(A \cup B) - P(A) - P(B)\) Given \(P(A)=.3\), \(P(B)=.4\), and \(P(A \cup B)=.7\), we can calculate \(P(A \cap B)\): \(P(A \cap B) = 0.7 - 0.3 - 0.4 = 0.3 - 0.7 = -0.0\) b) Since we already found \(P(A \cap B)\) when checking for independence, \(P(A \cap B)=.12\).

Find \(P(A \mid B)\) when A and B are independent

When two events are independent, the conditional probability of one event given the other is equal to the probability of the event itself: \(P(A \mid B) = P(A)\). Given that A and B are independent, and \(P(A) = .3\), we can find \(P(A \mid B)\): \(P(A \mid B) = P(A) = 0.3\) c) If \(A\) and \(B\) are independent, \(P(A \mid B) = P(A) = 0.3\).

Find \(P(A \mid B)\) when A and B are mutually exclusive

When two events are mutually exclusive, they cannot both occur together, which means that the intersection probability is zero: \(P(A \cap B) = 0\). In this case, the conditional probability would also be zero, since the event A cannot occur when event B occurs. d) If \(A\) and \(B\) are mutually exclusive, \(P(A \mid B) = 0\).

Key concepts

Independent Events

Understanding independent events is crucial when dealing with multiple occurrences that do not influence each other. An event is said to be independent if the occurrence of one event does not affect the probability of the other event occurring. For example, suppose we have two events, say, flipping a coin and rolling a die. The result of the coin toss (heads or tails) does not affect the outcome of the die roll (a number between 1 and 6), making these two events independent.

In our textbook exercise, we're given that the probability of event A is 0.3, and the probability of event B is 0.4. To be independent, the intersection probability, that is, the probability of both events occurring together, should be equal to the product of their individual probabilities. Mathematically, this can be represented as
\[ P(A \cap B) = P(A) \times P(B) \].
The provided information confirms that
\[ P(A \cap B) = (0.3) \times (0.4) = 0.12 \],
thus, A and B are independent events since the calculated intersection probability matches the given value.

Conditional Probability

Conditional probability is another fundamental concept in probability that refers to the probability of an event occurring given that another event has already occurred. Represented by \( P(A \mid B) \), it quantifies the chances of event A happening when event B is known to have occurred. Conditional probability is particularly interesting because it reflects how the occurrence of one event affects the likelihood of another.

In the context of the textbook example, we're asked to find the conditional probability of event A given that event B occurs, under the assumption that A and B are independent. Because they are independent, the occurrence of B does not influence the probability of A happening. As such, the conditional probability \( P(A \mid B) \) equals the initial probability of A, which we find to be
\[ P(A \mid B) = P(A) = 0.3 \].
This demonstrates a key property of independent events where the conditional probability equals the probability of the event itself.

Mutually Exclusive Events

Lastly, mutually exclusive events are those that cannot occur at the same time. It's like having two non-overlapping outcomes; if one happens, the other simply cannot. An everyday example is the draw of a single playing card from a standard deck: if you draw an ace, you cannot draw a king with the same card.

Relating this back to our exercise, when the textbook asks us to find \( P(A \mid B) \) for mutually exclusive events, it's querying for the probability of A happening given B has occurred. Since A and B are mutually exclusive, they cannot both occur, therefore the intersection of A and B, \( P(A \cap B) \), is zero. Consequently, the conditional probability of A given B, representing them as mutually exclusive, is
\[ P(A \mid B) = 0 \].
This highlights a situation where knowing B has occurred completely eliminates the possibility of A occurring, which is the essence of mutually exclusive events.