Probability Of Independent Events And Its Examples

The probability of independent events is the probability that one event does not affect the outcome. For example, a roulette game has many independent events. Each player has the same odds of winning in this game, but it’s impossible to know what will happen because each spin can be influenced by the last. It’s possible that one card in your hand will give you a straight and help you win big, or that you’ll draw two cards with big hands and abandon hope completely. The probability of independent events does not change unless there is an interaction between events.

Probability of independent events:

In probability theory, two events are said to be independent if the occurrence does not affect the others. 

For example, rolling a die twice is an experiment with four possible outcomes. A probability formula for this experiment is P= 16×2×16=136 If two dice are rolled, any single roll is considered an independent event from any other roll (that is P= 136 for each roll). If they are not independent, then probability of each roll depends on what happened with the previous roll, and this particular experiment must be examined in more detail.

In statistics, independence is generally assumed for statistical inference about parameters and the validity of tests based on them.

However, an important assumption in statistical inference and for the validity of tests is that the data are independent; if this is violated, it may result in misleading or inappropriate conclusions. For example, obtaining a p-value for a test of independence is meaningless unless there is no dependence between the data. This point is easily demonstrated with a non-random example. 

Consider the following series of coin tosses:

Heads: 1,1,1,1,1 … Tail: 1,2,3,4 … Heads: 2,3,3 … Tail: 4 ….. 

The result appears to be a series of “tails” followed by “heads” on successive throws. However, the “tails” are  simply a series of “heads” followed by a “tail.” The probability that these series have appeared in equal-length intervals is very small.

Independent events in probability formula:

Independent events in probability formula for two independent events, A and B:

P(A & B) = P(A)*P(B).

In the equation, we write “P(A)” for “the probability of A”, “P(B)” for “the probability of B”, and so on. The formula a friend showed me was:

In the formula above, we can see that [p(A) * p(B)] is equal to [1 – (1 – P(A))*(1 – P(B))]. Also remember that P(A) = 0.

Independent events examples:

Probability is the chance of occurrence of an event. Independent events are events that occur without affecting each other. Some independent event examples are as follows:

Example 1: Suppose you throw a die twice and get an ace every time. Is it an independent event?

Answer: No, as the ace coming up on dice is indirectly related to the first throw. It’s not a random event, independent of the first.

Example 2: The probability of getting an ace on a die is 0.25; what is the probability of getting a 1 and an ace together?

Answer: It is 0.25 + 0.01 = 0.

Independent events are only related to each other if they occur simultaneously. For example, in your dice throwing example, it is possible that both an ace and a 1 could come up simultaneously. However, even if this happens, it does not determine whether or not there is also a second die thrown.

Example 3: The probability of being immunized against polio in one month is 0.03. What is the probability that one of your three kids would be immunized by then?

Answer: 0.03/3 = 0.0001

Example 4: There are four cards in a deck. What is the probability of getting an ace on the first draw?

This is equal to 1/52×1/51×1/50×1/49=4.28%    

This is the probability of drawing an ace on the first draw if you know that the other three cards are all aces. It’s also equal to 1/52×1/51×1/50=4.28%     

Here’s another scenario: what is the probability of getting two aces and not getting any other card?

The answer is: 1/52×1/51=4.28%. This would be equal to (1 /252) = 0.000152.

We can calculate the probability of drawing one ace and anyone another card.

This is equal to 1/51×1/52-×1/51 = 4.28%.

What if we draw two cards? The answer would be 1/52×1/51×1/50=4.28%. We can generalize this result: the probability of drawing each card, as long as N cards are in the deck, is equal to (1/52)N.

Conclusion: 

The probability of independent events means the probability that an event does or does not affect another can be computed. The calculation is made by dividing the probability of both events independent of the sum of their probabilities.

Independent events are probably more numerous than many people think, but they are still comparatively rare. A series of coin tosses are independent because each toss has a 50% chance of landing on either heads or tails. However, unlike the roulette example above, the results of one coin toss will probably not affect the results of other tosses.