"In Exercises 1-4, identify the sample space of the probability experiment and determine the number of outcomes in the event. Draw a tree diagram when appropriate.
1. Experiment: Tossing four coins
Event: Getting three heads"

Verified step by step guidance

1. Understand the problem: The experiment involves tossing four coins, and the event of interest is getting exactly three heads. Each coin toss has two possible outcomes: heads (H) or tails (T).
2. Identify the sample space: The sample space consists of all possible outcomes of tossing four coins. Each outcome is a sequence of four letters (H or T), representing the result of each coin toss. For example, one possible outcome is HHHT (three heads and one tail).
3. Determine the total number of outcomes in the sample space: Since each coin toss has two outcomes and there are four coins, the total number of outcomes is given by \( 2^4 \).
4. Count the outcomes for the event: To get exactly three heads, we need to select three positions for heads out of the four coin tosses. This can be calculated using the combination formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), where \( n = 4 \) and \( r = 3 \).
5. Draw a tree diagram: Start with the first coin toss (H or T), then branch out for the second coin toss (H or T), and continue branching for the third and fourth tosses. Highlight the paths that result in exactly three heads to visualize the event.

Key Concepts

Sample Space

The sample space is the set of all possible outcomes of a probability experiment. In the case of tossing four coins, the sample space includes every combination of heads and tails that can occur, which can be represented as a list of outcomes such as HHHH, HHHT, HHTH, and so on. Understanding the sample space is crucial for calculating probabilities and analyzing events.

Event

An event is a specific outcome or a set of outcomes from the sample space that we are interested in. For the experiment of tossing four coins, the event of getting three heads includes outcomes like HHHT, HHTH, HTHH, and THHH. Identifying the event helps in determining the number of favorable outcomes for calculating probabilities.

Tree Diagram

A tree diagram is a visual representation used to illustrate all possible outcomes of a probability experiment. Each branch of the tree represents a possible outcome at each stage of the experiment. For tossing four coins, a tree diagram can help visualize the combinations of heads and tails, making it easier to count the total outcomes and identify specific events.