Question

State whether the two events (A and B) described are disjoint, independent, and/or complements. (It is possible that the two events fall into more than one of the three categories, or none of them.) Roll two (six-sided) dice. Let \(A\) be the event that the first die is a 3 and \(B\) be the event that the sum of the two dice is 8

Short answer

The two events A (the first die rolling a 3) and B (rolling a total sum of 8) are neither disjoint, independent, nor complements.

Step-by-step solution

Define the Events

Firstly, it's important to define and understand the events. Event A is when the first die rolls a 3. Thus there are 6 possible outcomes for event A, corresponding to the 6 different results that the second die can land on. Event B is when the sum of the two dice equals 8. The possible outcomes for event B are {(2,6), (3,5), (4,4), (5,3), (6,2)}. This gives a total of 5 outcomes.

Analyze for Disjoint Events

Disjoint events are events that cannot happen simultaneously. This is checked by observing if outcomes of the first event can exist in the second event at the same time. Here, one of the outcomes of event B, which is (3, 5), involves rolling a 3 on the first die, which is event A. Thus, events A and B are not disjoint.

Analyze for Independent Events

Two events are independent if the occurrence of one event does not affect the probability of the occurrence of the other event. The probability of Event A is 1/6 (only one number out of 6 possibilities), and the probability of Event B is 5/36 (5 favorable outcomes out of total 36). The intersection of A and B is one event: {(3,5)}. The probability of its occurrence is 1/36 (1 favorable outcome out of total 36). If A and B were independent, the probability of their intersection should be the product of their individual probabilities. However, 1/36 is not equal to (1/6) * (5/36), so A and B are not independent.

Analyze for Complementary Events

Two events are complements if one event occurs whenever the other does not. In this case, neither event A nor event B guarantees the non-occurrence of the other, so A and B are not complementary events.

Key concepts

Disjoint Events

When we talk about disjoint events in probability, we're referring to events that cannot occur at the same time. Think of them as mutually exclusive outcomes. For example, if you flip a coin, it can't land on both heads and tails simultaneously – these outcomes are disjoint. In the context of rolling two dice, if we define Event A as rolling a 3 on the first die, any outcome that includes a 3 on the first die belongs exclusively to Event A.

To determine disjoint events, we examine the potential outcomes. The outcome (3,5) is included in Event B where the sum equals 8, meaning that Event A, rolling a 3, and Event B can indeed happen together. This tells us that Events A and B are not disjoint, since there's overlap in at least one possible result where the first die is a 3 and the sum is 8.

Independent Events

Independent events are essentially the 'loners' of probability. The occurrence of one event doesn't influence the likelihood of the other occurring. Flipping a coin and rolling a die are classic independent events because getting heads on the coin has no impact on rolling a 6 on the die.

For events A and B with dice, to test for independence, we investigate if the probability of their combined occurrence (intersection) is the same as the product of their individual probabilities. The probability of rolling a 3 (Event A) is 1/6, and the probability of the two dice totaling 8 (Event B) is 5/36. However, the combined event of A and B occurring simultaneously (rolling a 3 and totaling 8) has a probability of 1/36. Since 1/36 is not equal to the multiplication of the probabilities of A and B (1/6 * 5/36), we conclude that these events are not independent – one event affects the probability of the other.

Complementary Events

Complementary events are like two sides of the same coin, literally! In probability terms, one event is the complement of another if one and only one of them will occur. This implies that the two events cover all possible outcomes together, and their probabilities add up to one.

For our dice-rolling scenario, we'd be looking to see if occurring of Event A (rolling a 3 on the first die) would automatically mean that Event B (sum of dice being 8) does not occur, and vice versa. It turns out this is not the case. Rolling a 3 (Event A) doesn't necessarily prevent the sum from being 8 – they can happen together, as seen in the outcome (3,5). Therefore, we determine that Event A is not the complement of Event B, and they are not complementary events.

Probability Intersection

The intersection of probability, represented by the symbol \( \cap \), is a fancy way of saying 'both at the same time.' When two events intersect, it means they're happening simultaneously. In gambling movies, this might be the protagonist rolling a pair of sixes when they most need it – that's an intersection.

Looking at our dice example, the intersection of Event A (the first die is a 3) and Event B (the sum is 8) would require the outcomes to satisfy both conditions. The only outcome that does this is (3,5) – rolling a 3 on the first die and having a total of 8. The intersection of these two events is therefore one occurrence, out of all 36 possible outcomes when two dice are rolled, leading to a probability of 1/36 for their intersection.