Problem 28 - Algebra 2

Question

Tami, Sonia, Malik, and Roger are the four candidates for Student Council president. If their names are placed in random order on the ballot, what is the probability that Malik's name will be first on the ballot followed by Sonia's name second?

Step-by-Step Solution

Answer

The probability is \(\frac{1}{12}\).

Step 1: Determine Total Possible Outcomes

Calculate the total number of ways to arrange the four names Tami, Sonia, Malik, and Roger in order. This is a permutation of four items, which can be calculated as \(4!\).

Step 2: Compute 4!

The calculation for the permutation is \(4! = 4 \times 3 \times 2 \times 1 = 24\). There are 24 possible ways to arrange the four names.

Step 3: Find Desired Outcomes

Determine the number of favorable outcomes where Malik's name is first and Sonia's name is second. With Malik fixed in the first position and Sonia in the second, we need to arrange Tami and Roger in the remaining two positions.

Step 4: Compute Permutations for Remaining Names

The remaining Tami and Roger can be arranged in 2! ways, which is \(2! = 2 \times 1 = 2\). Thus, there are 2 favorable outcomes where Malik and Sonia occupy the first two positions.

Step 5: Calculate Probability

The probability of Malik being first and Sonia being second is the number of favorable outcomes divided by the total possible outcomes: \(\text{Probability} = \frac{2}{24} = \frac{1}{12}\).

Key Concepts

Permutations

Permutations play a crucial role in understanding the arrangement of items in different sequences. A permutation is the way we can order a set of distinct items. In this exercise, we deal with the permutation of four names: Tami, Sonia, Malik, and Roger. To find the total arrangements possible for these four individuals, we use the concept of permutations.We calculate this by finding the factorial of the number of items, denoted as \(n!\). For our four candidates, the calculation would be \(4!\). This represents all the different ways you can line them up, considering that each order is distinct.

Favorable Outcomes

When solving problems with probability, identifying the favorable outcomes is essential. These are the specific instances that satisfy the conditions described in a problem. In our exercise, the favorable outcome is when Malik's name appears first on the ballot, and Sonia's name appears second. To determine this, we fix these positions and then count how many ways the remaining names can be arranged. By focusing on the desired condition, favorable outcomes help in computing probabilities accurately.

Factorial

The factorial function is a key part of permutations and combinations. It provides a way to calculate the number of arrangements of a set of items.The notation \(n!\) refers to the product of all positive integers up to \(n\). For instance, \(4!\) means multiplying 4 by 3, 2, and 1, or \(4 \times 3 \times 2 \times 1 = 24\). This is how we find the total number of permutations for our candidates.By understanding the factorial function, we can easily calculate different kinds of arrangements and comprehend their complexities.

Desired Outcomes

In probability exercises, desired outcomes refer to the specific scenarios that align with the conditions set out in the problem. They are a subset of the total possible outcomes.For Malik to be first and Sonia to be second, we once again use the factorial concept but now with the remaining names. After setting Malik and Sonia in their desired positions, we find the different ways to order Tami and Roger, which results in \(2!\) or 2 ways. This signifies the desired outcomes for this specific arrangement.Desired outcomes are critical as they directly measure the probability we're trying to determine in these types of problems.