8.2 Basic Statistics & Probability

8.2 Basic Statistics & Probability

Basic statistics and probability are key quantitative topics tested in the Saudi General Aptitude Test (GAT / Qudurat). These concepts assess your capability to summarize datasets using statistical measures—such as mean, median, mode, and range—and to analyze the likelihood of events under various conditions. The questions do not require university-level statistics but focus on core properties and algebraic applications of averages and probability rules.

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Measures of Central Tendency

Statistical measures summarize a set of data points into single representative values. The GAT tests four primary measures:

1. The Arithmetic Mean (Average)

The mean is the sum of all terms in a dataset divided by the total number of terms:

$\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}$

In GAT questions, you are often given the mean and must find a missing value or calculate a new mean when values are added. The most powerful way to solve these is to reorganize the formula to focus on the sum:

$\text{Sum} = \text{Mean} \times \text{Number of values}$

Worked Example: A student scored 80, 85, 88, and 92 on four mathematics quizzes. What score must they obtain on their fifth quiz to raise their average score to 87?

1. Calculate the target sum for 5 quizzes to achieve an average of 87: $\text{Target Sum} = 87 \times 5 = 435$
2. Calculate the current sum of the first 4 quizzes: $\text{Current Sum} = 80 + 85 + 88 + 92 = 345$
3. Determine the required score for the fifth quiz: $\text{Fifth Quiz Score} = \text{Target Sum} - \text{Current Sum} = 435 - 345 = 90$

Note on Averages of Consecutive Integers: If a set consists of consecutive integers (or any arithmetic progression like consecutive even numbers), the mean is simply the middle number (for an odd number of terms) or the average of the two middle numbers (for an even number of terms). It is also equal to the average of the first and last terms: $\frac{\text{First} + \text{Last}}{2}$.

2. The Median

The median is the middle value of a dataset when the numbers are arranged in ascending or descending order.

• Odd number of terms: The median is the single middle number. For a list of $n$ numbers, it is at position $\frac{n+1}{2}$.
• Even number of terms: The median is the average of the two middle numbers. For a list of $n$ numbers, these are at positions $\frac{n}{2}$ and $\frac{n}{2} + 1$.

Worked Example: Find the median of the following set: ${14, 5, 18, 21, 9, 11}$.

1. Arrange the set in ascending order: ${5, 9, 11, 14, 18, 21}$.
2. Since there are 6 terms (an even number), locate the two middle terms: 11 and 14.
3. Find their average: $\frac{11 + 14}{2} = 12.5$. The median is 12.5.

3. The Mode

The mode is the value that appears most frequently in a dataset. A dataset can have one mode, multiple modes (if two or more values share the highest frequency), or no mode (if all values appear only once).

4. The Range

The range measures the spread of the dataset and is calculated as the difference between the maximum and minimum values:

$\text{Range} = \text{Maximum Value} - \text{Minimum Value}$

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Probability Principles

Probability measures the likelihood of a specific event occurring, expressed as a number between 0 (impossible) and 1 (certain).

$\text{Probability } P(E) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}$

1. Independent Events

Two events are independent if the occurrence of one does not affect the probability of the other. The probability of both events occurring is the product of their individual probabilities:

$P(A \text{ and } B) = P(A) \times P(B)$

Worked Example: A fair six-sided die is rolled, and a coin is tossed. What is the probability of rolling a 4 and flipping heads?

• Probability of rolling a 4: $P(\text{4}) = \frac{1}{6}$
• Probability of flipping heads: $P(\text{heads}) = \frac{1}{2}$
• Combined probability: $\frac{1}{6} \times \frac{1}{2} = \frac{1}{12}$

2. Dependent Events (Without Replacement)

Two events are dependent if the outcome of the first event changes the probability of the second. This commonly occurs in problems involving selecting items from a bag "without replacement."

Worked Example: A box contains 5 green marbles and 3 yellow marbles. If two marbles are drawn at random one after another without replacement, what is the probability that both marbles are green?

1. The probability that the first marble is green is $\frac{5}{8}$ (5 green out of 8 total).
2. Since the first marble was green and not replaced, there are now 4 green marbles left and 7 total marbles left in the box.
3. The probability that the second marble is green is $\frac{4}{7}$.
4. Multiply the two probabilities: $\frac{5}{8} \times \frac{4}{7} = \frac{20}{56} = \frac{5}{14}$.

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Counting Principles

The GAT often tests your ability to count the total number of possible configurations or arrangements.

The Fundamental Counting Principle

If one task can be completed in $m$ different ways, and a second task can be completed in $n$ different ways, then the two tasks together can be completed in $m \times n$ different ways.

Worked Example: A restaurant offers a meal combo consisting of one main course, one drink, and one dessert. If there are 5 main courses, 3 drinks, and 4 desserts available, how many different combos can be created? $\text{Total Combos} = 5 \times 3 \times 4 = 60$

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GAT Traps and Test-Taking Strategies

1. The Unsorted Median Trap

A favorite trap on the GAT is to ask for the median of a list of numbers that are written in random order. Students in a hurry will pick the number in the physical middle of the list. Always arrange the list in order first!

2. The Average of Averages Trap

If Group A has an average score of 80 and Group B has an average score of 90, the combined average is NOT necessarily 85. It is only 85 if both groups have the exact same number of people. If Group A has 10 people and Group B has 30 people, you must find the total sum of scores first:

• Sum A: $10 \times 80 = 800$
• Sum B: $30 \times 90 = 2700$
• Combined Sum: $800 + 2700 = 3500$
• Total People: $10 + 30 = 40$
• Combined Average: $\frac{3500}{40} = 87.5$

3. Probability Denominator Decrement

In multi-stage probability questions, always check if the item is replaced or not. If it is not replaced, remember to reduce both the numerator (favorable outcomes) and the denominator (total outcomes) for subsequent selections.