What is Probability?
Probability refers to the chance or likelihood of a particular event taking place. It’s a numerical measure between 0 and 1 that expresses how probable an event is. The closer a probability is to 1, the more certain the event. Conversely, a probability closer to 0 indicates a less likely event.
What is an experiment?
An experiment is process that is performed to understand and observe possible outcomes. for example, rolling a dice is an experiment.
What is sample space?
Set of all outcomes of an experiment is called the sample space. for example, while rolling a dice the event may be 1, 2, 3, 4, 5 or 6, in this case the set of number from 1 to 6 is the sample space as these are all the possible outcomes.
What are favorable outcomes?
An outcome which same as the desired outcome is favorable outcome. for example, If getting a heads in a coin flip is desired outcome and the result of the flipping is also heads, it is a favorable outcome.
What is an event?
An event is an outcome of the experiment. for example, getting a number between 1 to 6 while rolling a dice is an event.
there are different types of events,
- Mutually exclusive events
- Independent events
- Dependent events
What are mutually exclusive events?
Mutually exclusive events are the events that cannot happens at the same time and exactly one of them must happen, these are also referred as complimentary events, for example, war and peace are mutually exclusive events they cannot happen at the same time.
What are independent events?
Independent events are the events whose occurrence is not dependent on any other event, for example, while flipping a coin event of heads or tails are not dependent on each other. So, these events are independent events.
What are dependent events?
Dependent events are the events in an experiment that are affected by other event, for example, drawing a marble from a bag which has marbles 2 red and 3 green marbles. In first event when you drew 1 red marble now in the second event the probability of red or green marble is affected by the first event. Because now the size of sample space has changed.
What are equally likely events?
The events are said to be equally likely if the chances of them occurring are same. We can understand this with coin flipping example, here the chances for a heads or tails are same, hence they are equally likely.
What are complimentary events?
In a series of events, if an event can only occur when other event did not occur. These dependent events are referred to as complimentary events.
What are simple events?
When probability is being calculated for a single event, it is called simple event. for example, tossing a single coin.
What are compound events?
When probability is being calculated for a more than one events at the same time, it is called compound event. for example, tossing more than one coin at a time.
How to calculate probability?
Probability of events
The basic formula for calculating probability is:
P(E) = Favorable Outcomes / Total Possible Outcomes
where:
- P(E) represents the probability of event E occurring.
- Favorable Outcomes are the number of ways the desired event can happen.
- Total Possible Outcomes represent all the possible outcomes of the experiment or situation.
Example
Imagine flipping a fair (unbiased) coin. There are two possible outcomes: heads or tails. Let’s calculate the probability of getting heads:
- Favorable Outcomes: 1 (getting heads)
- Total Possible Outcomes: 2 (heads or tails)
Therefore, the probability of getting heads (P(heads)) is:
P(heads) = 1 / 2 = 0.5
This means there’s a 50% chance of getting heads when flipping a fair coin.
Rules of computing Probability
Complementary Events
- The probability of the complement of event E (denoted by E’) is:
P(E') = 1 - P(E)
Addition Rule (for Disjoint Events)
- The probability of events in mutually exclusive events is calculated by the below formula,
P(A U B) = P(A) + P(B)
This rule applies to events which are mutually exclusive meaning they cannot occur together, these are also referred as disjoint, here the probability of either A or B happening is denoted by (A U B).
Addition Rule (for Overlapping Events)
- The probability of events in non-mutually exclusive events is
P(A U B) = P(A) + P(B) - P(A∩B)
When two events can occur together, or we can say when two events are overlapping, to avoid the overcounting outcomes we need to adjust the outcomes that fall into both categories. Here, P(A∩B) represents the probability of both A and B happening together (intersection of events)
Multiplication Rule (for Independent Events)
- The probability of two independent events occurring together,
P(A∩B) = P(A) * P(B)
When two events are independent of each other, what is the probability of combination of outcomes, for example, choosing a movie genre and snack. if you had to choose between comedy, drama and action and for snack you had to choose between popcorn and candy.
Event A: Picking a comedy movie (P(A) = 1/3, assuming you are equally likely to pick comedy, drama, or action)
Event B: Selecting popcorn as your snack (P(B) = 1/2, since you could choose popcorn or candy
What is the probability of selecting a comedy movie and popcorn? (P(A∩B))
Using the formula: P(A∩B) = P(A) * P(B) = (1/3) * (1/2) = 1/6
Conditional Probability
- The probability of second event after first event
P(A|B)=P(A∩B)/P(B)
- P(A|B) represents the probability of event A happening given that event B has already happened.
- P(A∩B) directly represents the number of elements (outcomes) that belong to both sets A and B.
- P(B) represents the total number of elements (outcomes) within set B.
Let’s take the example of train being delayed due to rainy weather, given that it is raining (Event A), what is the probability of train being delayed or we can say how much does the rain increases the chance of training getting delayed?
Here is a table summarizing an example scenario, focusing on weather conditions that might cause delays:
| Weather Condition | Total Days with Condition | Delayed Trains on Those Days |
|---|---|---|
| Rain | 20 | 10 |
| Snowfall | 10 | 8 |
| Fog | 15 | 5 |
| Total | 45 | 23 |
What is the probability of train getting delayed due to any weather?
- Total Delayed Trains = sum of delays across all weather conditions = 10 (rainy) + 8 (snowfall) + 5 (fog) = 23
- Total Days with Any Weather Condition = 45
- Therefore, Probability (Train Delayed due to any weather) = Total Delayed Trains / Total Days with Any Weather Condition = 23 / 45 ≈ 0.51 (or 51%)
What is the probability of train getting delayed due to different weather conditions?
- P(Train Delayed | Rain) = Favorable Delays on Rainy Days / Total Rainy Days = 10 / 20 = 0.5 (50%)
- P(Train Delayed | Snowfall) = Favorable Delays on Snowy Days / Total Snowy Days = 8 / 10 = 0.8 (80%)
- P(Train Delayed | Fog) = Favorable Delays on Foggy Days / Total Foggy Days = 5 / 15 = 0.33 (33%)
This analysis shows that chances of the train getting delayed are highest on rainy days
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Reference: Khan Acedemy