# Simple Probability

#### Grade 7 Math Worksheets

**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

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## Simple Probability - Grade 7 Math Worksheet PDF

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## 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.

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## 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%.

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## Simple Probability FAQS

##### What is probability?

Probability is a measure of the likelihood that an event will occur. It is expressed as a number between 0 and 1, with 0 indicating that the event is impossible and 1 indicating that the event is certain.

##### What is the difference between theoretical probability and experimental probability?

Theoretical probability is calculated based on the assumption of an idealized model, while experimental probability is based on actual observations or experiments. Theoretical probability is often used in mathematical models and simulations, while experimental probability is used to estimate real-world probabilities.

##### What is the law of large numbers?

The law of large numbers is a statistical principle that states that as the number of trials or observations increases, the average of the results will approach the expected value. This principle is important in probability theory and statistics because it helps to establish the reliability of estimates based on a large sample size.

##### What is the difference between independent and dependent events?

Independent events are events that have no effect on each other. The occurrence of one event does not affect the probability of the other event. Dependent events, on the other hand, are events that are affected by each other. The occurrence of one event affects the probability of the other event.

##### What is Bayes' theorem?

Bayes’ theorem is a formula used to calculate conditional probabilities. It is based on the idea that the probability of an event A given event B can be calculated by multiplying the probability of event B given event A by the prior probability of event A and dividing by the probability of event B.

##### What is a random variable?

A random variable is a variable that takes on different values depending on the outcome of a random event. It is often used in probability theory to represent the possible outcomes of an experiment or simulation.

##### What is a probability distribution?

A probability distribution is a function that describes the likelihood of different outcomes in a random event. It is often used in probability theory to model the possible outcomes of an experiment or simulation.

##### What is the difference between discrete and continuous probability distributions?

Discrete probability distributions are used to model events with a finite or countable number of possible outcomes, such as flipping a coin or rolling a die. Continuous probability distributions are used to model events with an infinite number of possible outcomes, such as measuring the height of a person or the amount of time it takes for a car to stop.

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