7. Probability

● Probability is a measure of how likely an event is to occur. ● It ranges between 0 (impossible event) and 1 (certain event).

Probability is a numerical value representing the likelihood of a certain event happening. This value can vary between 0 and 1. If the probability is 0, it means the event cannot happen at all, whereas a probability of 1 means that the event is certain to happen. For instance, when flipping a fair coin, the chances of getting heads or tails are each 0.5, indicating that both outcomes are equally likely.

Consider a simple example of a weather forecast. If the forecast states there is a 0% chance of rain, we can confidently say it will not rain (impossible event). On the other hand, if it states there is a 100% chance of rain, we can be sure that it will rain (certain event). The probability values directly inform us about the likelihood of various outcomes in daily situations.

● Experiment: An action that produces outcomes (e.g., tossing a coin). ● Trial: Performing the experiment once. ● Outcome: The result of a trial (e.g., heads or tails). ● Sample Space (S): Set of all possible outcomes. ● Event (E): A subset of the sample space.

✦ Example: When a die is rolled, what is the sample space? Solution: Sample space = {1, 2, 3, 4, 5, 6}

To understand probability, we need to know some basic terms. An 'experiment' is any procedure that can be repeated and has a set of possible results. A 'trial' refers to one instance of this experiment. The 'outcome' is the result from a trial. The 'sample space' is the collection of all outcomes. An 'event' is a specific result or a subset of the outcomes from the sample space. For example, if we roll a die, the sample space consists of {1, 2, 3, 4, 5, 6}, as these are all possible outcomes.

Think of an experiment as baking a cake. Each time you bake one, that's a trial. The various ways the cake can turn out (delicious, burnt, undercooked) are the outcomes. The sample space would be all the different possible cakes you could make, while an event might be just the delicious cakes from that variety.

✦ Explanation: If all outcomes of an experiment are equally likely, then: P(E)=Number of favorable outcomesTotal number of outcomes P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} ✦ Example: Find the probability of getting a tail when a coin is tossed. Solution: Favorable outcomes = 1 (tail) Total outcomes = 2 (head, tail) P(tail)=12P(tail)=\frac{1}{2}

In classical or theoretical probability, we assume that all outcomes of an experiment are equally likely, which simplifies calculating probabilities. The probability of an event happening can be calculated using the formula P(E) = (Number of favorable outcomes) / (Total number of outcomes). For example, when flipping a coin, there are 2 possible outcomes (heads or tails), and since one of those (tails) is favorable, the probability is 1 favorable outcome out of 2 possible outcomes, or 1/2.

Imagine you are at a birthday party with a straightforward game where children can either choose a toy car or a toy doll from a basket containing one of each. The children each have an equally likely chance to pick either toy. If one child wants to know their chances of picking the car, they can calculate it as the number of toy cars divided by the total number of toys—making it clear and straightforward in understanding their odds of selection.

✦ Explanation: Simple events involve only one outcome from the sample space. ✦ Example: Find the probability of getting a number greater than 4 when a die is rolled. Solution: Numbers greater than 4 = {5, 6} → 2 favorable outcomes Total outcomes = 6 P(number > 4)=26=13P(number > 4)=\frac{2}{6} = \frac{1}{3}

A simple event is an event that consists of just one outcome from the sample space. To calculate the probability of a simple event, we count how many favorable outcomes correspond to that event and divide it by the total number of outcomes. For instance, when rolling a die, the sample space is {1, 2, 3, 4, 5, 6}. The event of interest is getting a number greater than 4, which includes outcomes 5 and 6. Thus, we have 2 favorable outcomes out of 6 total outcomes, giving us a probability of 2/6 or 1/3.

Consider a box of chocolates where you want to pick a chocolate that has caramel filling. If the box contains 6 chocolates (1 with caramel and 5 without), the event of picking a caramel chocolate is a simple event. There’s only one way to succeed (choosing the caramel chocolate) from a total of 6 chocolates, so your probability is 1 out of 6, making it easy for you to understand your chances at a glance.

✦ Explanation: If P(E) is the probability of an event E, then the probability of the event not happening is: P(not E)=1−P(E) ✦ Example: The probability of it raining today is 0.65. What is the probability that it will not rain? Solution: P(not raining)=1−0.65=0.35

Complementary events are outcomes that are mutually exclusive; if one happens, the other cannot occur. If you know the probability of an event E occurring, you can easily calculate the probability of it not occurring using the formula P(not E) = 1 − P(E). This means if the chance of rain today is 65% (0.65), then the chance of it not raining today is 1 - 0.65 = 0.35 or 35%. This gives a clear understanding that both possibilities cover all possible outcomes.

Think of flipping a coin. The outcome can either be heads or tails. If you know that the coin will show heads with a probability of 50% (0.5), then the probability of it not being heads (which is tails) must also be 50%. This clear dichotomy helps illustrate how complementary events work in real-world scenarios, making it easier to grasp their interdependence.

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