What is the probability of the spinner landing on 2

Probability Calculation

The probability of an event occurring can be calculated using the formula:

Finding Favorable Outcomes

In this scenario, the spinner is divided into 4 equal parts, each labeled 1, 2, 3, and 4. The favorable outcome here is landing on the number 2. Therefore, we have the following:

• Number of favorable outcomes: 1 (since there's only one sector labeled 2)
• Total number of possible outcomes: 4 (since there are 4 equal parts)

Calculating the Probability

To find the probability of the spinner landing on 2, we apply the values into the formula:

Where:

Thus, the probability of the spinner landing on 2 is 0.25 or 25%.

Explanation

The number of favorable outcomes refers to the count of specific outcomes of an event that correspond to the event of interest in a probability experiment. These are the outcomes that we consider successful or desired among all possible outcomes.

In probability theory, if we denote:

• $E$ as the event of interest
• $S$ as the sample space (the set of all possible outcomes)

Then, the number of favorable outcomes to $E$ is typically represented by $|E|$.

Calculation

To determine the probability $P(E)$ of an event $E$, we can use the formula:

Example

Consider a simple example: rolling a fair six-sided die.

• The sample space $S$ for a single roll of the die is:

  $$S = \{1, 2, 3, 4, 5, 6\}$$

• Suppose the event $E$ is to roll an even number:

  $$E = \{2, 4, 6\}$$

Here, the number of favorable outcomes $|E|$ is $3$ (since there are 3 even numbers), and the total number of possible outcomes $|S|$ is $6$.

The probability $P(E)$ of rolling an even number is then calculated as:

Key Points

• Sample Space $S$: The set of all possible outcomes.
• Event $E$: Subset of the sample space representing the outcome(s) of interest.
• Number of Favorable Outcomes $|E|$: Count of outcomes within event $E$.

Understanding the number of favorable outcomes is critical for solving many problems in probability theory. It helps quantify the likelihood of a particular event occurring within the given sample space.

Explanation

The "total number of possible outcomes" refers to the complete set of distinct results that can occur in an experiment or a probabilistic scenario. Understanding this concept is fundamental in statistics and probability, as it forms the basis for further calculations such as probabilities and expected values.

Counting Principle

One key method to determine the total number of possible outcomes is the Counting Principle. If an event can occur in $n$ different ways, and a subsequent event can occur in $m$ different ways, then the total number of combined outcomes for both events is $n \times m$.

Example: Rolling Dice

Consider the case of rolling two dice. Each die has 6 faces, so for one die, there are 6 possible outcomes. If we roll two dice, the total number of outcomes is given by:

General Formula

For a sequence of $k$ events, where each event $i$ has $n_i$ possible outcomes, the total number of outcomes is given by the product:

Combinatorics

In some scenarios, particularly when dealing with combinations and permutations, combinatorial methods are used to calculate the number of possible outcomes. For example:

• Permutations: The number of ways to arrange $n$ distinct objects is $n!$ (factorial of $n$).
• Combinations: The number of ways to choose $k$ objects from $n$ distinct objects (without regard to order) is given by the binomial coefficient:

Importance

Understanding the total number of possible outcomes is crucial for making informed decisions based on probability, conducting statistical analyses, and solving various real-world problems in fields such as genetics, game theory, and risk management.