01 · Explore
Understanding Probability and Randomness
Probability is a mathematical measurement of how likely an event is to occur, similar to how we measure physical quantities like length or volume.
In our daily lives, we encounter many situations where the outcome is uncertain, such as whether it will rain today or which team will win a match. These are called random events. While we may know all the possible outcomes, we cannot predict with 100% certainty which one will happen in a single instance.
Randomness refers to a situation or action, like tossing a coin or rolling a die, where the result is unpredictable. A random experiment is an observation that can be repeated where the result might be different every time. For example, in a cricket match, a coin toss is considered fair because both 'Heads' and 'Tails' have an equal chance of occurring, making the specific outcome unpredictable for both captains.
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Why is a coin toss considered a random experiment?
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02 · Explore
The Probability Scale
Probability is measured on a numerical scale from 0 to 1 to indicate the certainty of an event.
The probability of any event, denoted as P(E), always falls between 0 and 1 (0 ≤ P(E) ≤ 1). A probability of 0 means the event is impossible, such as rolling a 7 on a standard six-sided die. A probability of 1 means the event is certain, such as picking a red sweet from a bag containing only red sweets.
Values in between represent varying degrees of likelihood. A probability of 0.5 (or 50%) represents an 'even chance,' meaning the event is just as likely to happen as it is not to happen. As the value moves closer to 1, the event becomes 'more likely'; as it moves closer to 0, it becomes 'less likely'.
| Probability Value | Description | Example |
|---|---|---|
| 0 | Impossible | Getting a number > 6 on a standard die |
| Close to 0 | Less Likely | Rolling a specific number (like 3) on a die |
| 0.5 | Even Chance | Flipping a coin and getting Heads |
| Close to 1 | More Likely | Drawing a numbered card (2-10) from a deck |
| 1 | Certain | Choosing a red sweet from a bag of all red sweets |
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If an event has a probability of 0.75, what does this mean in terms of likelihood?
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03 · Explore
Experimental Probability
Experimental probability is based on actual data collected from performing trials or observing past events.
Experimental probability, also known as relative frequency, is calculated by dividing the number of times a specific outcome occurs by the total number of trials conducted. This method is used extensively in fields like business, insurance, and science where we rely on evidence from experience rather than just logic.
For example, if you toss a paper cup 100 times and it lands on its side 70 times, the experimental probability of it landing on its side is 70/100 or 0.7. This value is an estimate based on that specific set of trials.
Calculating Experimental Probability
Experimental Probability = (Number of times event occurred) / (Total number of trials)
Suppose you roll a die 50 times and it lands on the number 4 exactly 8 times. To find the experimental probability: 8 / 50 = 0.16. This can also be expressed as 16%.
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A student tosses a coin 20 times and gets 12 Heads. What is the experimental probability of getting Heads?
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04 · Explore
Theoretical Probability
Theoretical probability is what we expect to happen in an ideal, perfectly fair situation where all outcomes are equally likely.
Unlike experimental probability, theoretical probability does not require performing an experiment. It is based on reasoning. We assume the objects involved (like coins or dice) are 'unbiased' or 'fair,' meaning they are symmetrical and have no reason to land on one side more than another.
To calculate theoretical probability, we identify the number of 'favourable outcomes' (the results we are looking for) and divide by the 'total number of possible outcomes' in the sample space.
Probability of Picking a Letter
P(Event) = (Number of favourable outcomes) / (Total possible outcomes)
In the word 'PROBABILITY', there are 11 letters in total. If we want the probability of picking the letter 'B', we count how many 'B's are there (2). Thus, P(B) = 2/11 ≈ 0.182 or 18.2%.
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What is the theoretical probability of getting an even number on a fair 6-sided die?
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