Calculating Probability
Table Of Contents
🎬 Math Angel Video: Probability Explained Step by Step
What is Probability?
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🛎️ Definition of Probability:
Probability tells us how likely a certain event is to happen.
It helps us describe the chance of something occurring.
❇️ Key Idea:
Probabilities range from 0 to 1, or 0% to 100%.
The closer the probability is to 1 (100%), the more likely the event will happen.
| Probability | Description | Example |
|---|---|---|
| 0 (0%) | Impossible to happen | Rolling a 10 on a normal 6-sided die 🎲 |
| 0.25 (25%) | Unlikely to happen | Picking a red ball from a bag with mostly blue |
| 0.75 (75%) | Likely to happen | Picking a red ball from a bag with mostly red |
| 1 (100%) | Certain will happen | The sun rises in the east ☀️ |
How to Calculate Probability?
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Calculating probability means finding out how likely an event is to occur.
🛎️ What Is an Event?
Imagine you pick one ball from a bag of 10. 3 are red and 7 are blue.
- Outcome is one possible result. Each ball you pick is one outcome. There are 10 possible outcomes in total.
- Event is a group of outcomes you care about. For example, “picking a red ball” is an event; “picking a blue ball” is another event.
🛎️ What Is the Probability Formula?
$$\text{Probability of an event} = \frac{\text{Number of desired outcomes}}{\text{Total number of outcomes}}$$
Example: There are 10 balls in total:3 red and 7 blue. What is the probability of drawing a red ball at random?
$$P(\text{red}) = \frac{3}{10} = 0.3 = 30\%$$
So, there’s a 30% chance of picking a red ball.
What is Experimental Probability?
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Experimental probability is an estimate of how likely something is to happen, based on the results of a real experiment.
It shows what actually happens when an event is repeated many times, rather than what we expect to happen in theory.
🛎️ How to Find Experimental Probability?
Example: A bag contains red and blue balls, but we don’t know the exact ratio. To estimate the probability, a student draws one ball, notes its colour, puts it back, and repeats this process 100 times.
Results:
- Red balls drawn: 37 times
- Blue balls drawn: 63 times
$$P(\text{red}) = \frac{37}{100} = 0.37 = 37\%$$
So the experimental probability of drawing a red ball is 37%.
$$P(\text{blue}) = \frac{63}{100} = 0.63 = 63\%$$
So the experimental probability of drawing a blue ball is 63%.
🛎️ Compared with Theoretical Probability
If we knew there were 3 red and 7 blue balls in the bag, the theoretical probability would be:
$$P(\text{red}) = \frac{3}{10} = 0.3 = 30\%$$
But in the experiment, the actual result was 37%, slightly higher.
❇️ Key Ideas
Theoretical probability is what we expect to happen, based on known facts or logic.
Experimental probability is what we observe to happen, from real data.
The more trials we do, the closer experimental results get to the true probability.
(This is called the Law of Large Numbers.)
🍪 Practice: Probability Calculations and Experimental Probability
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