Problem 54 - Algebra 2

Question

Find each probability if a coin is tossed 4 times \(P(4 \text { heads })\)

Step-by-Step Solution

Answer

\( \frac{1}{16} \)

Step 1: Understand the Problem

We are tasked with finding the probability of getting 4 heads in 4 coin tosses. Each toss has two possible outcomes: heads or tails.

Step 2: Define the Total Number of Outcomes

For each coin toss, there are 2 possible outcomes: heads (H) or tails (T). When a coin is tossed 4 times, the total number of possible outcomes is calculated as \(2^4 = 16\).

Step 3: Identify the Favorable Outcomes

We are interested in the single outcome where we get 4 heads: HHHH. Thus, there is exactly 1 favorable outcome.

Step 4: Calculate the Probability

The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. Therefore, the probability of getting 4 heads is \(\frac{1}{16}\).

Key Concepts

Coin Toss Probability

Tossing a coin is a classic example of a probability exercise. Each coin flip is an event that can result in either two outcomes: heads or tails. When a coin is tossed multiple times, the number of possible outcomes grows exponentially.
For example, if you toss a coin four times, you have multiple sequences possible, such as HHTT, HTHT, etc. These sequences are what we term as the 'sample space' in probability. Understanding coin toss probability provides a foundation to grasp more complex probabilistic concepts and calculations that extend into different fields of study.

Probability Calculation

Probability is a measure of the likelihood that an event will occur. It is quantified between 0 and 1, where 0 indicates an impossible event and 1 indicates a certain event. In our coin toss example, you have:

• A total number of possible outcomes which is derived from the number of tosses and possible results per toss. Since each coin flip has 2 possible outcomes, for 4 coins, the total number is calculated using \(2^4\), giving 16 possible outcomes.
• The number of favorable outcomes, which in this case, is just 1 sequence: HHHH.

The probability is then calculated using the formula: \[P( ext{event}) = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}}\]So for getting 4 heads, \(P(4 \text{ heads}) = \frac{1}{16}\).

Favorable Outcomes

In probability language, a 'favorable outcome' is one that corresponds to the event you are trying to measure. These outcomes are defined by the conditions set in the problem.
Here, because we are focusing on getting 4 heads, HHHH is our only favorable outcome among the 16 sequences possible when tossing a coin four times. Having a clear understanding of what constitutes a favorable outcome is essential. It helps cut through the complexity of more intricate probability scenarios as you can isolate these outcomes to aid in accurate calculation.