Simple Probability

Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 means that the event is impossible and 1 means that the event is certain.

Table of Contents:

• Simple Probability
• Formula
• Examples
• FAQs

Simple Probability

Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 means that the event is impossible and 1 means that the event is certain.

To calculate the probability of an event, you need to divide the number of favorable outcomes by the total number of possible outcomes.

For example, if you roll a fair six-sided die, the probability of rolling a 1 is 1/6, because there is only one favorable outcome (rolling a 1) out of six possible outcomes (rolling a 1, 2, 3, 4, 5, or 6).

Another example: if you draw a card from a standard deck of 52 playing cards, the probability of drawing a spade is 13/52, because there are 13 spades in the deck (favorable outcomes) out of 52 cards in total (possible outcomes).

It is important to note that probabilities can be expressed as fractions, decimals, or percentages. For example, a probability of 1/2 can also be written as 0.5 or 50%.

Formulas of Probability

Here are some standard probability formulas:

Probability of an event A: P(A) = number of favorable outcomes for A / total number of possible outcomes

Probability of the complement of event A (not A): P(not A) = 1 – P(A)

Probability of the intersection of events A and B (both A and B occur): P(A and B) = P(A) * P(B|A)

Probability of the union of events A and B (either A or B or both occur): P(A or B) = P(A) + P(B) – P(A and B)

The conditional probability of event B given event A (the probability of B given that A has occurred): P(B|A) = P(A and B) / P(A)

Bayes’ theorem (used to calculate the probability of event A given event B): P(A|B) = P(B|A) * P(A) / P(B)

These formulas are used in various branches of mathematics and statistics, including probability theory, combinatorics, and inferential statistics.

Examples of Probablity

Sure, here are some examples of how the probability formulas can be applied:

Probability of an event A: A coin is flipped. What is the probability of getting heads? There is one favorable outcome (heads) out of two possible outcomes (heads or tails), so the probability of getting heads is 1/2.

Probability of the complement of event A: Using the same example as above, the probability of getting tails (not getting heads) is 1/2, because it is the complement of the event of getting heads.

Probability of the intersection of events A and B: A bag contains 4 red balls and 3 blue balls. Two balls are drawn at random from the bag without replacement. What is the probability of drawing a red ball first and then a blue ball?

The probability of drawing a red ball on the first draw is 4/7. Given that a red ball was drawn on the first draw, there are now 3 red balls and 3 blue balls left in the bag, so the probability of drawing a blue ball on the second draw is 3/6 (since one ball has already been removed). Therefore, the probability of drawing a red ball first and then a blue ball is (4/7) * (3/6) = 2/7.

Probability of the union of events A and B: Using the same example as above, what is the probability of drawing a red ball on the first draw or a blue ball on the second draw (or both)? We already calculated the probability of drawing a red ball first and then a blue ball as 2/7. The probability of drawing a blue ball first and then a red ball is the same, because the order of the draws doesn’t matter. So the probability of drawing a red ball first or a blue ball second is (4/7) * (3/6) + (3/7) * (4/6) = 4/7.

Conditional probability of event B given event A: A family has two children. You are told that at least one of them is a boy. What is the probability that both children are boys? The event A is “at least one boy”, and the event B is “both children are boys”. There are four possible outcomes for the gender of the two children: BB, BG, GB, and GG (where B stands for boy and G stands for girl). Since we know that at least one of the children is a boy, we can eliminate the GG outcome. That leaves three equally likely outcomes: BB, BG, and GB. Only one of those outcomes satisfies the event B (both children are boys). Therefore, the conditional probability of B given A is 1/3.

Bayes’ theorem: A certain disease affects 1% of the population. A test for the disease has a false positive rate of 5% (meaning that 5% of people without the disease will test positive) and a false negative rate of 10% (meaning that 10% of people with the disease will test negative). If a person tests positive, what is the probability that they actually have the disease?

Let A be the event that a person has the disease, and let B be the event that the person tests positive. We want to calculate P(A|B), the probability of having the disease given that the person tests positive. Using Bayes’ theorem, we can write:

P(A|B) = P(B|A) * P(A) / P(B)

To calculate the probability of having the disease given that the person tests positive (P(A|B)), we can use Bayes’ theorem. Bayes’ theorem states that:

P(A|B) = (P(B|A) * P(A)) / P(B)

where:

P(A|B) is the probability of having the disease given that the person tests positive,

P(B|A) is the probability of testing positive given that the person has the disease,

P(A) is the probability of having the disease,

P(B) is the probability of testing positive.

Let’s calculate each of these probabilities step by step:

1) P(A) = 0.01 (given that the disease affects 1% of the population)

2) P(B|A) = 1 – 0.10 = 0.90 (since the false negative rate is 10%, the probability of testing positive given that the person has the disease is 1 – 0.10 = 0.90)

3) P(B) = P(B|A) * P(A) + P(B|not A) * P(not A)

= 0.90 * 0.01 + 0.05 * 0.99

= 0.009 + 0.0495

= 0.0585

P(B|not A) = 0.05 (false positive rate, 5%)

P(not A) = 1 – P(A) = 1 – 0.01 = 0.99 (probability of not having the disease)

Now, let’s calculate P(A|B):

P(A|B) = (P(B|A) * P(A)) / P(B)

= (0.90 * 0.01) / 0.0585

= 0.009 / 0.0585

≈ 0.154

Therefore, the probability that a person actually has the disease given that they test positive is approximately 0.154, or 15.4%.

Simple Probability FAQS

Grade 7 Science Worksheets

• Elements and Compounds
• Solar Energy
• Photosynthesis
• Electricity and Magnetism
• Law of conservation of energy
• Periodic table
• Properties of Matter
• Waves
• Energy Resources 
• Weather and Climate
• Immune, Circulatory and Digestive Systems
• Organs in Multi-cellular Organism
• Sedimentary, Igneous, and Metamorphic Rocks
• Structure of the Earth
• Law of Conservation of Mass
• Physical and Chemical Changes
• Scientific Method
• Human Digestive System
• Environmental Science
• Renewable and Non-renewable energy Resources
• Characteristics of Living Organisms
• Life Science
• Earth and Space Science
• Solar Eclipse
• Heat Technology
• Newton’s Laws of Motions
• Physical Science
• Tools, Measurement and SI Units
• Earth Atmosphere
• Interactions of Living things
• The Earth Ecosystem
• Organelles in Plant and Animal cells
• Layers of the Earth
• Cycles in Nature

Grade 7 Math Worksheets

• Fractions
• Linear equations word problems
• Statistics
• Properties of Parallel Line
• Finding slope from an equation
• Identifying Quadrilaterals
• Percent Change
• Properties of addition and multiplication
• Pythagorean Theorem
• Solving two step inequalities
• Symmetry
• Fractions to Decimals (New)
• Whole Number Exponents with Integer Bases (New)
• Adding and Subtracting Fractions (New)
• Integer Addition and Subtraction (New)
• Dividing Mixed Numbers (New)
• Basics of Coordinate Geometry (New)