Quantitative Reasoning Study Guide for the DAT

Probability and Statistics

Probability and statistics are important components of mathematics, though they are sometimes the most dreaded by test-takers. Both fields are essential for making informed decisions based on data and understanding patterns and trends.

Probability

Probability is a measure of how likely an event is to occur, expressed as a number between \(0\) and \(1\). A probability of \(0\) indicates an impossible event such as the probability that the Sun will rise in the east, while a probability of \(1\) indicates a certain event such as the probability that a coin flip will result in either a head or a tail. In between, probabilities express varying degrees of likelihood. The probability of an event is denoted by \(p\), where \(0 \leq p \leq 1\), and it provides a numerical measure of the uncertainty or certainty associated with that event.

Terms

When dealing with probability, you will need to be familiar with the following terms:

• experiment—An experiment is a procedure that can be infinitely repeated and has a well-defined set of possible outcomes. For example, flipping a coin is an experiment where the possible outcomes are heads or tails.

• outcome—An outcome is a possible result of an experiment. For instance, when rolling a six-sided die, an outcome could be any of the numbers from \(1\) to \(6\).

• sample space—The sample space is the set of all possible outcomes of an experiment. For example, the sample space for rolling a die is \(\{1, \, 2, \, 3, \, 4, \, 5, \, 6 \}\).

• empty set—The empty set, denoted by \(\emptyset\), is a set with no elements. In the context of probability, it represents an impossible event. For instance, rolling a \(7\) on a standard six-sided die is not possible, so it is an event that corresponds to the empty set.

• independent event—Independent events are events in which the occurrence of one event does not affect the probability of the other event occurring. For example, flipping a coin twice is two independent events, because the result of the first flip won’t affect the result of the second flip.

• dependent event—Dependent events are events in which the occurrence of one event does affect the probability of the other event occurring. For example, drawing a card from a deck and not replacing it affects the probability of drawing another card.

AND/OR Probability

• AND probability (intersection)—The probability of both events A and B occurring is denoted as \(P(A \cap B)\). For independent events, \(P(A \cap B) = P(A) \times P(B)\).

For example, as mentioned above, flipping a coin twice is two independent events. Let \(A\) be the event that a first coin flip is heads and \(B\) be the event that a second coin flip is heads. Then, \(P(A) = P(B) = \frac{1}{2}\) and the probability that both coin flips is heads is:

\(P(A \cap B) = P(A) \times P(B) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\).

• OR probability (union)—The probability of either event A or event B occurring is denoted as \(P(A \cup B)\). For any events, \(P(A \cup B) = P(A) + P(B) - P (A \cap B)\).

For example, let us consider rolling a six-sided die. Let \(A\) be the event that a roll is an odd number and \(B\) be the event that a roll is a multiple of \(3\). Since there are six possible outcomes and three of them are odd numbers, \(P(A) = \frac{3}{6} = \frac{1}{2}\). On the other hand, there are two multiples of \(3\) on a six-sided die: \(3\) and \(6\). So, \(P(B) = \frac{2}{6} = \frac{1}{3}\). Finally, The only outcome that \(A\) and \(B\) have in common is \(3\). Therefore, \(P(A \cap B) = \frac{1}{6}\). Putting this all together, the probability that either event \(A\) or event \(B\) occur is:

\[P(A \cup B) = P(A) + P(B) - P(A \cap B)\] \[= \frac{1}{2} + \frac{1}{3} - \frac{1}{6}\] \[= \frac{3}{6} + \frac{2}{6} - \frac{1}{6}\] \[= \frac{3 + 2 - 1}{6}\] \[=\frac{4}{6} = \frac{2}{3}\]

Expressing Probability

Probability can be expressed as a fraction, decimal, or percentage. For example, if there is a one in four chance of an event occurring, it can be expressed as \(0.25\) (decimal) or \(25\%\) (percentage).

The basic probability formula is:

\[P(E) = \frac{n(E)}{n(S)}\]

where \(P(E)\) is the probability of an event, \(n(E)\) is the number of favorable outcomes, and \(n(S)\) is the total number of possible outcomes (or the sample space).

For example, the probability of rolling a \(3\) on a six-sided die is \(\frac{1}{6}\) since the number three is only one of a total six numbers.

One important aspect of probability calculation is the probability of an event not happening. The probability of an event not occurring is equal to \(1\) minus the probability of the event occurring:

\[P(\text{not } E) = 1 - P(E)\]

For example, if the probability of it raining today is \(0.3\), then the probability of it not raining is \(1 - 0.3 = 0.7\).

Statistics

Statistics is the branch of mathematics that deals with collecting, analyzing, interpreting, and presenting data. It provides tools for making sense of numerical information, identifying patterns, and drawing conclusions based on data sets. The following sections will cover essential concepts and measures in statistics, including mean, median, mode, range, standard deviation, combinations, and permutations.

Mean, Median, and Mode

The mean, or average, is calculated by adding all the values in a data set and dividing by the number of values. It represents the central tendency of the data. For the data set \(\{ 10, \, 16, \, 22, \, 18 \}\), the mean is \(\frac{10+16+22+18}{4} = \frac{66}{4} = 16.5\).

The median is the middle value of a data set when it is ordered from least to greatest. If there is an even number of values, the median is the average of the two middle numbers. For the data set \(\{ 11, \, 17, \, 12, \, 14, \, 9 \}\), the median is \(12\) because it is the middle number when we arrange the data set in ascending order (from smallest to largest). For the data set \(\{ 6, \, 12, \, 8, \, 20 \}\), we first arrange the numbers in ascending order:

\[\{ 6, \, 8, \, 12, \, 20 \}\]

Since there are an even number of values, we average the two middle values to get the median:

\[\frac{8+12}{2} = \frac{20}{2} = 10\]

The mode is the value that occurs most frequently in a data set. A data set can have more than one mode or no mode at all if no value repeats. For the data set \(\{ 1, \, 4, \, 5, \, 3, \, 4, \, 3, \, 4, \, 8, \, 3, \, 10, \, 3 \}\), the mode is \(3\) since it occurs the most number of times.

Range

The range is a measure of the spread of a data set. It is calculated by subtracting the smallest value from the largest value. See the data set below:

\[\{ 10, \, 34, \, 23, \, 12, \, 53, \, 32, \, 15 \}\]

The largest value is \(53\) and the smallest value is \(10\). Thus, the range is \(53 - 10 = 43\).

Standard Deviation

Standard deviation is a measure of the dispersion or spread of a data set. It indicates how much the values in the data set deviate from the mean. The formula for standard deviation is:

\[\sigma = \sqrt{\frac{\Sigma (x_i - \mu)^{2}}{N}}\]

where \(\sigma\) is the standard deviation, \(x_i\) represents each value in the data set, \(\mu\) is the mean, and \(N\) is the number of values.

Standard deviation measures the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values are close to the mean, while a high standard deviation indicates a wide range of values.

The formula can seem intimidating at first, but follow the steps below to calculate standard deviation:

1. Calculate the mean, \(\mu\), of the data set by dividing the sum of the values (\(S\)) by the number of values.
2. Subtract the mean from each value to find the deviation for each value.
3. Square each deviation.
4. Sum all squared deviations.
5. Divide by the number of values, \(N\), for population), or by \(N-1\), for sample.
6. Take the square root of the result.

For example, suppose we are trying to find the standard deviation of the following sample of a data set:

\[\{0, 1, 3, 5, 5, 5, 9\}\]

First, we calculate the mean:

\[\mu = \frac{S}{N} = \frac{0 + 1 + 3 + 5 + 5 + 5 + 9}{7} = \frac{28}{7} = 4\]

Next, we subtract the mean from each value:

\[0 - 4 = -4, \; 1 - 4 = -3, \; 3 - 4 = -1, \; 5 - 4 = 1, \; 5 - 4 = 1, \; 5 - 4 = 1, \; 9 - 4 = 5\]

Now, we square each deviation:

\[(-4)^2 = 16, \; (-3)^2 = 9, \; (-1)^2 = 1, \; 1^2 = 1, \; 1^2 = 1, \; 1^2 = 1, \; 5^2 = 25\]

Next, we sum these squares:

\[16 + 9 + 1 + 1 + 1 + 1 + 25 = 54\]

Now, since this is a sample, we divide this sum by the number of values minus 1:

\[\frac{54}{7 - 1} = \frac{54}{6} = 9\]

Finally, we take the square root of the result to get the standard deviation of our sample:

\[\sqrt{9} = 3\]

Combinations

Combinations refer to the selection of items from a larger pool where the order does not matter. The formula for combinations is:

\[C(n,k) = \frac{n!}{k! (n-k)!}\]

where \(n\) is the total number of items, \(k\) is the number of items to choose, and \(!\) denotes a factorial. For example, \(5!\) can be written as \(5 \times 4 \times 3 \times 2 \times 1\).

Use combinations when the order of selection does not matter. Suppose you want to find the possible combination of two fruits from a set of three fruits (mango, orange, and apple). You will do the following:

\[C(3,2) = \frac{3!}{2!(3-2)!} = \frac{3!}{2! \times 1!} = \frac{3 \times 2 \times 1}{2 \times 1} = 3\]

So, there are three combinations: apple and orange, apple and mango, and orange and mango.

Note: You can cancel out common factors in factorials to simplify problems. So, for instance, in \(\frac{3 \times 2 \times 1}{2 \times 1}\), the \(2\) and \(1\) in the numerator and denominator cancel each other out, leaving just \(3\).

Permutations

Permutations refer to the arrangement of items where the order does matter. The formula for permutations is:

\[P(n,k) = \frac{n!}{(n-k)!}\]

where \(n\) is the total number of items and \(k\) is the number of items to arrange.

Use permutations when the order of selection matters. If you wanted to arrange two fruits from a set of three fruits (mango, orange, and apple), you would do the following:

\[P(3,2) = \frac{3!}{(3-2)!} = \frac{3!}{1!} = \frac{3 \times 2 \times 1}{1} = 6\]

So, there are six ways you can arrange the fruit:

• mango, orange
• orange, mango
• mango, apple
• apple, mango
• orange, apple
• apple, orange

Geometry

Geometry is the branch of mathematics that studies the sizes, shapes, properties, and dimensions of objects and spaces. It is a foundational subject in mathematics and is essential for understanding and solving problems related to shapes and spatial relationships. Geometry can be divided into three parts: plane, solid, and coordinate.

Plane Geometry

Plane geometry is concerned with flat shapes that lie on a plane, a flat, two-dimensional surface. It includes the study of lines, angles, polygons (such as triangles and rectangles), and circles. Plane geometry helps us understand and describe the properties and relations of these shapes.

Lines and Points

Lines are one of the simplest units in plane geometry, right after points.

A line is a straight one-dimensional figure that extends infinitely in both directions. It has no thickness and is defined by any two points that lie on it. A point is a distinct location on a plane with no shape or size; it’s represented by a dot.

In the above figure, line \(AB\) extends infinitely in both directions, passing through points \(A\) and \(B\).

A line segment is part of a line that is bounded by two distinct endpoints. It contains all the points on the line between its endpoints.

The above figure shows line segment \(AB\), which has endpoints \(A\) and \(B\) and includes all points between \(A\) and \(B\).

A ray starts at one point and extends infinitely in one direction. It has a fixed starting point and no endpoint.

The above figure shows a ray starting at point \(A\) and passing through point \(B\), extending infinitely beyond \(B\) in the same direction.

Triangles

A triangle is a three-sided polygon. Triangles form the basis of many geometric concepts and properties.

Basics

A triangle has three sides and three angles. The sum of the interior angles of a triangle is always \(180\) degrees. This property is essential for solving many geometric problems. The length of the sides and measure of the angles determine the type of triangle. We will look at these types below.

Angles

A triangle has two types of angles: interior and exterior.

Remember that the sum of the three interior angles of a triangle is always \(180\) degrees. This rule is used to find unknown angles in triangles. For example, if a triangle is known to have two angles that are \(50^{\circ}\) and \(60^{\circ}\), we can find the third angle (\(x\)) in the following way:

\[50 + 60 + x = 180\] \[110 + x = 180\] \[x = 180 - 110\] \[x = 70^{\circ}\]

An exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles, the two angles inside the triangle that are not adjacent to the exterior angle. This property is useful in solving for unknown angles in more complex problems. For example, in a triangle, if one exterior angle is \(120\) degrees and one of the remote interior angles is \(40\) degrees, the other remote interior angle is:

\[120 - 40 = 80^{\circ}\]

Types

Based on the sides, we can classify triangles in the following three ways:

• scalene triangle—a triangle with all three sides of different lengths

• isosceles triangle—a triangle with two sides of equal length

• equilateral triangle—a triangle with all three sides of equal length

Based on the angles, we can also classify triangles in the following three ways:

• acute triangle—a triangle in which all three angles are less than \(90\) degrees

• obtuse triangle—a triangle in which one angle is greater than \(90\) degrees

• right triangle—a triangle with one \(90\)-degree angle

Rules

For every triangle, you can calculate its perimeter and area.

The perimeter (\(P\)) of a triangle is the sum of the lengths of its three sides. It represents the total distance around the triangle. The formula is:

\[P = a + b + c\]

where \(a\), \(b\), and \(c\) are the lengths of the three sides of the triangle.

The area (\(A\)) of a triangle can be calculated using the formula:

\[A = \frac{1}{2} \times b \times h\]

where \(b\) is the length of the base of the triangle and \(h\) is the height of the triangle.

One important rule for triangles is the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. This theorem ensures the possibility of constructing a triangle with given side lengths.

\[a + b>c\text{ and }a+c >b\text{ and }b+c>a\]

Note: \(a\), \(b\), and \(c\) are the three side lengths of a triangle.

Pythagorean Theorem

The Pythagorean theorem applies to right triangles and is a very useful principle in geometry. It states that the square of the hypotenuse, the side opposite the right angle, is equal to the sum of the squares of the other two sides:

\[a^2 + b^2 = c^2\]

For example, if the two legs of a right triangle are \(3\) and \(4\) inches, then the hypotenuse is:

\[3^2 + 4^2 = c^2\] \[25 = c^2\] \[c= \sqrt{25} = 5\text{ in}\]

These numbers, \(3\), \(4\), and \(5\), are known as pythagorean triplets. Some common pythagorean triplets you should know are:

\[(3,4,5), (5,12,13), (8,15,17), (7,24,25),\text { and } (9,40,41)\]

To solve a few triangle problems quickly and efficiently, you should also be familiar with two special right triangles:

• 45-45-90 triangle—In this triangle, the legs are equal, and the hypotenuse is \(\sqrt{2}\) times the length of each leg. If each leg is \(1\), the hypotenuse is \(\sqrt{2}\).

• 30-60-90 triangle—In this triangle, the lengths of the sides are in the ratio \(1:\sqrt{3}:2\). The side opposite the \(30^{\circ}\) angle is the shortest, the side opposite the \(60^{\circ}\) angle is \(\sqrt{3}\) times the shortest side, and the hypotenuse is twice the shortest side. So, if the shortest side is \(1\), the side opposite the \(60^{\circ}\) angle is \(\sqrt{3}\), and the hypotenuse is \(2\).