Problem 8 - Algebra and Trigonometry

Question

An experiment consists of tossing a coin and rolling a die. (a) Find the sample space. (b) Find the probability of getting heads and an even number. (c) Find the probability of getting heads and a number greater than \(4 .\) (d) Find the probability of getting tails and an odd number.

Step-by-Step Solution

Answer

(a) Sample space: {(H,1), (H,2), ..., (T,6)} (b) Probability: \(\frac{1}{4}\) (c) Probability: \(\frac{1}{6}\) (d) Probability: \(\frac{1}{4}\)

Step 1: Understanding the Sample Space

The sample space is the set of all possible outcomes of the experiment. We have two events here: tossing a coin and rolling a die. The outcomes for tossing a coin are Heads (H) or Tails (T). The outcomes for rolling a die are 1, 2, 3, 4, 5, 6. To find the sample space, combine each coin outcome with each die outcome.

Step 2: Constructing the Sample Space

The sample space consists of all the possible pairs of a coin result and a die result: \[ S = \{ (H,1), (H,2), (H,3), (H,4), (H,5), (H,6), (T,1), (T,2), (T,3), (T,4), (T,5), (T,6) \} \]This sample space has 12 outcomes.

Step 3: Calculating Probability of Heads and Even Number

To find this probability, first identify the favorable outcomes from the sample space where the coin shows heads and the die shows an even number. The even numbers on a die are 2, 4, and 6, so the favorable outcomes are \((H,2), (H,4), (H,6)\). There are 3 such favorable outcomes.

Step 4: Probability of Heads and Even

The probability is the number of favorable outcomes divided by the total number of outcomes in the sample space. Thus, \[ P(\text{Heads and even}) = \frac{3}{12} = \frac{1}{4} \].

Step 5: Calculating Probability of Heads and Number > 4

Identify the favorable outcomes for a result showing heads and a number greater than 4 on the die. These outcomes are \((H,5), (H,6)\). There are 2 such favorable outcomes.

Step 6: Probability of Heads and Greater Than Four

Similarly, the probability is given by:\[ P(\text{Heads and number} > 4) = \frac{2}{12} = \frac{1}{6} \].

Step 7: Calculating Probability of Tails and Odd Number

Consider the favorable outcomes where the coin shows tails and the die shows an odd number. The odd numbers on a die are 1, 3, and 5, so the favorable outcomes are \((T,1), (T,3), (T,5)\). There are 3 favorable outcomes.

Step 8: Probability of Tails and Odd

The probability is calculated as follows:\[ P(\text{Tails and odd}) = \frac{3}{12} = \frac{1}{4} \].

Key Concepts

Sample Space

In probability, the term "sample space" refers to the set of all possible outcomes of an experiment. In our scenario, we are conducting two activities: tossing a coin and rolling a die.

For the coin toss, there are two possible outcomes: Heads (H) or Tails (T). For the die roll, since a standard die is used, there are six outcomes possible: 1 through 6.

To construct the sample space for this compound event, we combine each outcome of the coin toss with each outcome from the die roll. Here is what the sample space will look like:

• (H,1)
• (H,2)
• (H,3)
• (H,4)
• (H,5)
• (H,6)
• (T,1)
• (T,2)
• (T,3)
• (T,4)
• (T,5)
• (T,6)

This leads to a total of 12 different possible outcomes, considering every combination of the two events.

Coin Toss

A coin toss is a simple probability experiment often used to make decisions due to its 50/50 nature. When tossing a coin, there are always two possible outcomes: It will either land on Heads (H) or Tails (T).

This inherent simplicity of a coin toss makes it an ideal candidate for probability exercises, serving as a basic building block for more complex statistical calculations.

Because the outcomes are both equally likely, the probability of getting either Heads or Tails in a single coin toss is \( \frac{1}{2} \). However, when combined with other events, such as rolling a die, this probability needs to be considered as part of a larger sample space.

Die Roll

A die roll adds more complexity compared to a coin toss because there are six possible outcomes. When you roll a standard die, you can get any number from 1 to 6. We call each of these numbers a possible outcome.

The probability of rolling a particular number on a fair die is \( \frac{1}{6} \), as each number has an equal chance of appearing.

In our case, each die roll is paired with the outcome of a coin toss to form a composite event. This expands the simple probability of the die roll into part of a larger, more complex sample space used to calculate probabilities for various outcomes.

Even and Odd Probabilities

The concepts of even and odd numbers come into play when dealing with probabilities involving a die. It’s important to first understand what makes a number even or odd.

{

• Even Numbers: Divisible by 2 (e.g., 2, 4, 6)
• Odd Numbers: Not divisible by 2 (e.g., 1, 3, 5)

} For a die, we have three even numbers and three odd numbers among its faces. Therefore, if you roll a die, the probability of landing on an even number is \( \frac{3}{6} = \frac{1}{2} \), which is the same as that of landing on an odd number.

When evaluating probabilities, if we consider the outcomes of a coin toss alongside a die roll, as in our initial scenario, these probabilities become part of the sample space and specific to each configuration, such as landing on "Heads" and an even number, "Tails" and an odd number, and so on.