Problem 3 - A First Course in Probability

Question

Two dice are thrown. Let \(E\) be the event that the sum of the dice is odd, let \(F\) be the event that at least one of the dice lands on \(1,\) and let \(G\) be the event that the sum is 5. Describe the events \(E F, E \cup F, F G, E F^{c},\) and \(E F G\).

Step-by-Step Solution

Answer

Event EF: Pairs of dice rolls - (1,2), (1,4), (1,6), (2,1), (4,1), and (6,1). Event E∪F: Pairs of dice rolls - (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,3), (2,5), (3,1), (3,2), (3,4), (3,6), (4,1), (4,3), (4,5), (5,1), (5,2), (5,4), (5,6), (6,1), (6,3), and (6,5). Event FG: Pairs of dice rolls - (1,4) and (4,1). Event E\(F^{c}\): Pairs of dice rolls - (2,3), (2,5), (3,2), (3,4), (3,6), (4,3), (4,5), (5,2), (5,4), (5,6), (6,3), and (6,5). Event EFG: Empty set, as no pair of dice rolls satisfy all three conditions simultaneously.

Step 1: Event E (Sum is Odd)

First, let's identify the possible outcomes for Event E - when the sum of the dice is odd. Since the dice have faces from 1 to 6, the possible sums can range from 2 to 12. The odd sums are 3, 5, 7, 9, and 11. Pairs of dice rolls resulting in even E: (1,2), (1,4), (1,6), (2,1), (2,3), (2,5), (3,2), (3,4), (3,6), (4,1), (4,3), (4,5), (5,2), (5,4), (5,6), (6,1), (6,3), and (6,5).

Step 2: Event F (At least one dice lands on 1)

Now let's identify the possible outcomes for Event F - when at least one of the dice lands on 1. Pairs of dice rolls resulting in event F: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (3,1), (4,1), (5,1), and (6,1).

Step 3: Event G (Sum is 5)

Finally, let's identify the possible outcomes for Event G - when the sum of the dice is 5. Pairs of dice rolls resulting in even G: (1,4), (2,3), (3,2), and (4,1). Now, let's describe the events based on the combinations of E, F, and G.

Step 4: Event EF (Intersection of E and F)

Event EF represents the intersection of events E and F. It will include the pairs where both the sum is odd and at least one dice lands on 1. Pairs of dice rolls for event EF: (1,2), (1,4), (1,6), (2,1), (4,1), and (6,1).

Step 5: Event E∪F (Union of E and F)

Event E∪F represents the union of events E and F. It will include the pairs where either the sum is odd or at least one dice lands on 1 (or both). Pairs of dice rolls for event E∪F: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,3), (2,5), (3,1), (3,2), (3,4), (3,6), (4,1), (4,3), (4,5), (5,1), (5,2), (5,4), (5,6), (6,1), (6,3), and (6,5).

Step 6: Event FG (Intersection of F and G)

Event FG represents the intersection of events F and G. It will include the pairs where the sum is 5 and at least one dice lands on 1. Pairs of dice rolls for event FG: (1,4) and (4,1).

Step 7: Event E\(F^{c}\) (Intersection of E and complement of F)

Event E\(F^{c}\) represents the intersection of event E and the complement of event F. It will include the pairs where the sum is odd and both dice are not 1. Pairs of dice rolls for event E\(F^{c}\): (2,3), (2,5), (3,2), (3,4), (3,6), (4,3), (4,5), (5,2), (5,4), (5,6), (6,3), and (6,5).

Step 8: Event EFG (Intersection of E, F, and G)

Event EFG represents the intersection of events E, F, and G. It will include the pairs where the sum is odd, at least one dice lands on 1, and the sum is 5 (all conditions must be satisfied simultaneously). Since no pair of dice rolls satisfy all three conditions simultaneously, event EFG is an empty set.

Key Concepts

Dice Games

Playing games with dice is a fun way to explore probability. When we roll a standard die, we can get any of six faces showing up. This randomness makes dice games a great example for understanding probability. In most games, two six-sided dice, each showing numbers from 1 to 6, are rolled to determine the outcome.

When two dice are thrown, the possible outcomes are all the combinations of the numbers each die can show. This results in a total of 36 possible outcomes, calculated as 6 outcomes from the first die multiplied by 6 from the second die. Dice games can be unpredictable but understanding probability makes the unpredictability just a bit more predictable. Knowing the likelihood of each possible outcome helps to strategize and make informed decisions during the game.

Event Intersection

In probability theory, the intersection of two events is an important concept. The intersection, denoted by \(E \cap F\), refers to the outcomes that are common to both events. This means only the outcomes that satisfy both conditions or characteristics of the two events are included.

For example, when rolling two dice:

• Let's say event E is having an odd sum.
• Event F is rolling at least one die showing a 1.

Event \(E \cap F\) consists of outcomes that have both an odd sum and include at least one die that rolled a 1. The key to finding intersections is to list down the outcomes of each event separately, then identify where these lists overlap.

Event Union

The union of two events in probability theory, represented by \(E \cup F\), refers to the set of outcomes that belong to either one or both of the events. When we discuss unions, we are talking about combining all the outcomes without duplicating those that occur in both events.

Imagine two dice again:

• If event E is that the sum is odd,
• and event F is that at least one die shows a 1.

Event \(E \cup F\) includes all outcomes from both conditions: odd sums and rolls showing a 1. Therefore, any combination exceeding both criteria just needs to satisfy one to be included in the union.

Probability Theory

Probability theory is the branch of mathematics that deals with the likelihood of different outcomes occurring. This theory helps us understand the concept of chance or the likelihood of events. When we discuss probability, we are often referring to measures such as:

• Variance
• Mean
• Standard deviation

These help describe the spread and central tendency of probability distributions.

In the context of dice, it involves determining how likely it is to roll a specific number or combination of numbers. For example, to calculate the probability of rolling a nine with two dice, you need to count the outcomes where the sum is 9 (like 3+6, 4+5), and divide it by the total outcomes. Understanding probability provides insights into everyday random processes and helps make calculated predictions about likely outcomes.