Probability-Simple Problems. Conditional Probability

Probability, simply put, is the chance of whether a specific event will occur or not. Based on the number of possible outcomes, the probability of a specific event predicts how favorable that event can be or what is the chance of having that event. Many events are impossible to predict with 100% accuracy, but the likelihood of their taking place may be predicted. All probabilities are expressed as a number between zero and one.

Conditional probability

The conditional probability formula is defined by the multiplication of the preceding event’s probability by the conditional or succeeding event’s probability. Basically, conditional probability focuses on the occurrence of a single event’s probability depending on the preceding event’s probability. 

Independence 

Two events are said to be independent of one another if the likelihood of one event occurring has no bearing on the likelihood of the other event occurring.

Conditional independence 

The conditional independence probability theory talks about the situations where an observation is completely redundant or unrelated during the evaluation of a hypothesis’ certainty.

Example

Let us consider ‘a’ as the given hypothesis where ‘b’ and ‘c’ are observations. In this scenario, conditional independence will be represented as: P(a|b,c) = P(a|c). Here, P(a|b,c) defines the probability of ‘a’ given both ‘b’ and ‘c.’ As the probability of a given ‘c’ has the similar probability as ‘a’ given both ‘b’ and ‘c.’ The equality obtained here shows that ‘b’ has no role in determining the certainty of ‘a.’ In this scenario ‘a’ and ‘b’ will be conditionally independent of given ‘c.’ It will be demonstrated as: (a || b|c). 

The nature of the third event determines conditional independence. When two dice are rolled, one might presume that they will behave independently of one another. Looking at the outcome of one die will not reveal the outcome of the other. (In other words, the two dice are unrelated.)

Distribution of probabilities

It is a mathematical function that translates the probabilities of all conceivable outcomes of a random experiment. Whether the random variable X is discrete or continuous is determined by the random variable X.

Example

Let us consider the tossing of a coin. Here, we take x as a random variable that represents the heads’ count that is obtained. So, here we will find the distribution of probability for x. 

Considering x’s values like 0, 1, and 2. We can take possibilities such as: 

• No number of headcounts

• One tail and one head

• Both coins have the heads

Here, we can describe distribution of probability as following: 

P(X=0) = P(Tail+Tail) = ½ * ½ = ¼

P(X=1) = P(Head+Tail) or P(Tail+Head) = ½ * ½ + ½ *½ = ½

P(X=2) = P(Head+Head) = ½ * ½ = ¼

Conditional probability distribution 

The conditional probability distribution of Y given X is the probability distribution of Y when X is known to be a certain value in probability theory and statistics. In some circumstances, the conditional probabilities may be written as functions containing the undetermined value x of X a parameter. A conditional probability table is often used to illustrate the conditional probability when both X and Y are categorical variables. The conditional distribution of a random variable differs from its marginal distribution, which is its distribution without regard for the value of the other variable.

The conditional density function is the probability density function of the conditional distribution of Y given X if it is a continuous distribution. The conditional mean and conditional variance are two terms used to describe the attributes of a conditional distribution, such as the moments.

Formula

For all x, use the formula p(x) = P(X=x). The probability mass function is the discrete random variable X.

Examples

Conditional probability: p(A|B) is the chance of event A happening if event B happens. What is the likelihood that you drew a red card (p(four|red))=2/26=1/13 if you drew a red card? So there are two fours out of the 26 red cards (given a red card), so 2/26=1/13.

Conclusion 

People employ probability and odds in various scenarios, such as deciding how to dress for the weather, whether or not to buy a stock, and how much to risk when gambling. Probability assists people in determining which options are safe and which are risky. Of course, having a thorough understanding of probability makes this process much easier. We can learn about the likelihood of future events and plan accordingly by knowing about probability.

Probability assists people in determining which options are safe and which are risky. Of course, having a thorough understanding of probability makes this process much easier. We can learn about the likelihood of future events and plan accordingly by knowing about probability.