Basic Probability Concepts

Introduction to probability

Probability is a measure of the likelihood of an event occurring. It is a mathematical concept used to estimate the likelihood of an event occurring. Probability helps us make decisions and forecast future events. Mathematically, probability is defined as a numerical statement of how likely an event is to occur or how likely a proposition is to be true. It ranges from 0 to 1, where 0 represents impossibility and 1 represents certainty.

Sample locations and events

Before delving deep into probability, let us understand the terms sample space and event.

Sample space

The sample space of an experiment is the set of all possible outcomes. For example, the sample space for a coin toss is {heads, tails} and for a dice toss it is {1, 2, 3, 4, 5, 6}.

Sample space for a coin flip: {Head, Tail}

Sample space for rolling a die: {1, 2, 3, 4, 5, 6}

Events

An event is a subset of the sample space. It is the set of outcomes of an experiment that correspond to a particular outcome or combination of outcomes. In the case of throwing a dice, if you want to find the probability of getting an even number, the event could be {2, 4, 6}.

Event of rolling an even number: {2, 4, 6}

Probability of an event

The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. The formula is:

P(Event) = Number of Favorable Outcomes / Total Number of Possible Outcomes

Let us consider an example where we need to find the probability of getting a 3 on a standard six-sided die:

• Number of favourable outcomes: 1 (as there is only one '3' on the dice)
• Total number of possible outcomes: 6 (since there are six faces)

The probability is this:

P(rolling a 3) = 1 / 6

Understanding probability with an example

Imagine a spinner divided into four equal parts: red, green, blue and yellow.

If you spin the spinner, what are the chances of landing on red?

• Number of favorable outcomes: 1
• Total number of possible outcomes: 4 (red, green, blue, yellow)

The probability is this:

P(landing on Red) = 1 / 4 = 0.25

Theoretical versus experimental probability

Probability can be expressed in two forms: theoretical probability and experimental probability.

Theoretical probability

Theoretical probability is determined by the possible outcomes of an event, assuming that each outcome has an equal chance of occurring. It is based on logic or calculations rather than experiments.

For example, the theoretical probability of getting heads when tossing a coin is 1/2, because there are two possible outcomes (heads and tails), and each is equally likely to occur.

P(Theoretical, landing on Heads) = 1 / 2

Experimental probability

Experimental probability is determined by conducting experiments or surveys and collecting data on actual outcomes. This form of probability is based on observation rather than calculation.

For example, if you toss a coin 100 times and it comes up heads 48 times, the experimental probability of coming up heads is:

P(Experimental, landing on Heads) = 48 / 100 = 0.48

Note that the experimental probability may differ from the theoretical probability due to the limited number of trials and randomness.

Independent and dependent events

Events can be classified as independent or dependent, based on whether their outcomes affect one another.

Independent events

Two events are considered independent if the occurrence of one event does not affect the probability of the occurrence of the other. For example, throwing a dice and tossing a coin are independent events because the outcome of throwing the dice does not affect the outcome of tossing a coin.

P(A and B) = P(A) × P(B)

Dependent events

Two events are considered dependent if the occurrence of one event affects the probability of the other event occurring. For example, drawing a card from a deck without replacement is a dependent event because the outcome of drawing the first card affects the outcome of drawing the second card.

P(A and B) = P(A) × P(B|A)

where P(B|A) is the probability of event B occurring given event A occurring.

Complementary programs

Complementary events are pairs of events where one event occurs only if the other event does not occur. The sum of the probabilities of complementary events is always 1.

For example, in tossing a coin, the outcome is either heads or tails. Getting heads in a coin toss and not getting heads (which is tails) in a coin toss are complementary events.

P(A) + P(Not A) = 1

where P(A) is the probability of event A, and P(Not A) is the probability of event A not occurring.

Mutually exclusive and non-mutually exclusive events

Events can also be classified as mutually exclusive or non-exclusive, depending on whether they can occur simultaneously.

Mutually exclusive events

Mutually exclusive events cannot happen at the same time. For example, when a dice is thrown, the events of getting a 2 and a 5 are mutually exclusive because they cannot happen simultaneously.

P(A or B) = P(A) + P(B)

Non-mutually exclusive events

Non-mutually exclusive events can occur at the same time. For example, in a standard deck of cards, drawing a red card and drawing a heart can occur simultaneously because hearts are red cards.

P(A or B) = P(A) + P(B) - P(A and B)

Where P(A and B) is the probability of both events A and B occurring simultaneously.

Conclusion

In short, probability is a fundamental concept that measures the likelihood of events occurring. It is essential for making predictions and decisions in everyday life. Understanding basic probability concepts such as sample space, events, independent and dependent events, complementary events, and mutually exclusive events equips us with the knowledge to tackle a wide range of problems statistically.

As you progress in learning probability, these foundational concepts form the basis for more advanced topics and applications in various fields including finance, science, engineering, etc.

Comments