Probability and statistics with SAT Practice Test #1 | SAT Suite of Assessments – SAT Help 24×7

Probability - SAT HELP 24x7

Probability is a measure of the likelihood of an event happening. It's usually expressed as a fraction or a decimal between 0 and 1, where:

• \( P = 0 \) means the event is impossible and will not happen.
• \( P = 1 \) means the event is certain and will happen.

The basic formula for probability is:

\[ P(E) = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}} \]

Example:

Imagine you have a standard six-sided die. What's the probability of rolling a 3?

There's 1 side with a 3 and 6 sides in total, so:

\[ P(3) = \frac{1}{6} \]

Statistics

Statistics is a branch of mathematics that deals with data collection, analysis, interpretation, and presentation. Let's start with some basic concepts.

Mean (Average):

The mean is the sum of all the numbers divided by the count of numbers.

\[ \text{Mean} = \frac{\text{Sum of all numbers}}{\text{Count of numbers}} \]

Median:

The median is the middle value in a list of numbers. To find the median:

1. Put the numbers in numerical order.
2. If there's an odd number of values, the median is the middle number.
3. If there's an even number of values, the median is the average of the two middle numbers.

Mode:

The mode is the number that appears most frequently in a list.

Example:

Given the set of numbers: 2, 3, 4, 4, 5, 5, 5, 6

• Mean: \[ \text{Mean} = \frac{2+3+4+4+5+5+5+6}{8} \]
• Median: Since there are 8 numbers (even), the median is the average of the 4th and 5th numbers: \( \frac{4+5}{2} \)
• Mode: The number 5 appears most frequently (3 times), so the mode is 5.

Based on our calculations:

• The Mean (Average) of the numbers is \(4.25\).
• The Median of the numbers is \(4.5\).

Range:

The range is the difference between the highest and lowest numbers in a set. For the given set of numbers:

Range = Highest number - Lowest number

The Range of the numbers is \(4\).

Variability:

Variability (or spread) describes how much individual numbers in a dataset differ from the mean. The most common measures of variability are:

1. Variance: It's the average of the squared differences from the mean.
2. Standard Deviation: It's the square root of the variance. It gives a sense of how spread out the values in a dataset are around the mean.

Formula for variance (\( \sigma^2 \)):

\[ \sigma^2 = \frac{\sum_{i=1}^{n} (x_i - \text{mean})^2}{n} \]

Where:

• \( x_i \) represents each value in the dataset.
• \( \text{mean} \) is the mean of the dataset.
• \( n \) is the number of values in the dataset.

Standard Deviation is simply the square root of the variance (\( \sigma \)):

\[ \sigma = \sqrt{\sigma^2} \]

For our dataset:

• The Variance is \(1.4375\).
• The Standard Deviation is approximately \(1.199\).

Probability Distributions:

A probability distribution describes how the values of a random variable are distributed. It shows the probabilities of different outcomes. For the SAT, you'll primarily come across discrete distributions (like the binomial distribution) rather than continuous distributions.

Binomial Distribution:

This is a discrete probability distribution of the number of successes in a fixed number of independent Bernoulli trials (with only two possible outcomes, usually termed success and failure).

The probability of getting exactly \( k \) successes in \( n \) trials is:

\[ P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \]

Where:

• \( \binom{n}{k} \) is the binomial coefficient (number of ways to choose \( k \) successes from \( n \) trials).
• \( p \) is the probability of success on a given trial.

Example:

Imagine you flip a fair coin 3 times. What's the probability you get exactly 2 heads?

Here:

• \( n = 3 \) (3 trials)
• \( k = 2 \) (2 successes)
• \( p = 0.5 \) (since the coin is fair)

The probability of getting exactly 2 heads when flipping a fair coin 3 times is \(0.375\) or \(37.5\%\).

Summary:

We've covered some basic concepts in probability and statistics, including:

• The fundamentals of probability
• Measures of central tendency: mean, median, and mode
• Measures of variability: range, variance, and standard deviation
• An introduction to probability distributions, specifically the binomial distribution