# Probability Calculator

## Probability of Two Events

Probability is a way to gauge how likely something is to happen. It is expressed as a number between 0 and 1, with 1 denoting certainty and 0 representing the impossibility of the event. It follows that the likelihood of an event increasing increases with probability, making its occurrence more likely. Probability can be expressed mathematically in the most generic sense as the ratio of desired outcomes to total outcomes. This is further influenced, among other things, by whether the events being investigated are independent, mutually exclusive, or conditional. Using the given calculator, you can determine the likelihood that an event A or B will not happen, the probability that A or B will occur when they are not mutually exclusive, the possibility that both A and B will happen, and the likelihood that either A or B will happen but not both.

## Use of Probability Calculators

Calculate the likelihood of the outcome and the percentage odds of winning and losing using the supplied odds. With the help of this calculator, you may convert "odds for winning" or "odds against winning" an event into percentage chances of both outcomes.

Use betting odds or odds for sports teams with caution. If you find that the Patriots' super bowl odds are 9/2, those are most likely "odds against", and you should write "Odds are: against winning" in the calculator.

Ensure you comprehend the chances or probability stated by the game's organiser before participating in a lottery or other game of chance. This calculator is set to "1 to 500 Odds are for winning," representing a chance of 1 in 500. You may also see the probability of winning stated as 500:1. This most certainly indicates that "1 to 500 Odds are for winning" is the same as "500 to 1 Odds are against winning."

**Statistical formulas:**

- With the help of this calculator, you may convert an event's "odds of winning" into a likelihood percentage chance of succeeding.
- Probabilities are expressed as (chances for success): (chances against success) or the opposite.
- The probability of winning is expressed as PW = A / (A + B) if the odds are given as an A to B chance of winning, and the probability of losing is given as PL = B / (A + B).
- You lose if you pull any other card. For instance, you win the game if you draw an ace from a full 52-card deck. The odds of winning are 4 out of 52, and the odds of losing are 48 out of 52 (52-4=48). A=4 and B=48 are entered into the calculator since the winning chances are 4:48.
- 4 to 48 per cent chance of winning;
- Winnability is (0.0769) or 7.6923 per cent likely.

losing 92.3077 per cent (0.9231) - ""Winning: 1:12 (down from 4:48)" odds
- chances against "victory: 12:1 (reduced from 48:4)

## Combination of A and B

It is easy to determine the compliment, or the likelihood that the event indicated by P(A) does not occur, P(A'), given a probability A, denoted by P(A). If, for instance, P(A) = 0.65 indicates the likelihood that Bob will not complete his schoolwork, his teacher Sally can forecast the likelihood that Bob will do it as follows:

**P(A') = 1 - P(A) = 1 - 0.65 = 0.35**

Therefore, there is a 35% likelihood that Bob will complete his assignment given this scenario. Any P(B') would be computed in the same way. It's important to note that in the calculator above, P(B') can be independent, meaning that if P(A) = 0.65, P(B') doesn't necessarily have to equal 0.35 but might also equal 0.30 or another value.

### Complement of A and B

The combined probability of at least two occurrences is represented below in a Venn diagram as the intersection of events A and B, also written as P(A B) or P(A AND B). P(A B) = 0 when A and B are occurrences that cannot co-occur. Take into account the impossibility of rolling a 4 and a 6 on the same die. Therefore, these occurrences would be thought to be mutually exclusive. If the circumstances are independent, computing P(A B) is easy. The likelihood of events A and B in this situation is doubled. To determine the probability that two different die rolls result in 6 each time:

The supplied calculator considers the scenario in which the probability is unrelated. When two events are interdependent, calculating the likelihood requires knowledge of conditional probability, often known as the probability of event A, given that event B has occurred, or P(A|B). Consider a bag of ten marbles, seven of which are black and three of which are blue. Determine the likelihood of drawing a black marble if a blue marble has been removed from the bag without being replaced (lowering the total number of marbles in the bag):

The likelihood of drawing a blue marble

**P(A) = 3/10**

The likelihood of drawing a black marble

**P(B) = 7/10**

Chance of drawing a black marble after drawing a blue marble:

**P(B|A) = 7/9**

As can be observed, any previous instance in which a black or blue marble was drawn without a replacement impacts the likelihood that a black marble will be removed. To ascertain the possibility of taking a blue and then a black marble from the bag:

Using the above probabilities, what are the odds of drawing a blue marble first, then a black marble?

**P(A B) = P(A B|A) = (3/10) (7/9) = 0.2333**

## Intersection of A and B

There are two scenarios for the union of events: either the occurrences are mutually exclusive or they are not. Keep in mind that P(A U B) is another way to write P. (A OR B). The "inclusive OR" is applied in this situation. This means that all of the union conditions may be true simultaneously, provided that at least one of them does. The calculation of the probability is more straightforward when the circumstances are mutually exclusive:

The rolling of a die, where event A represents the likelihood that an even number will be rolled and event B represents the likelihood that an odd number will be moved, serves as a simple illustration of mutually exclusive occurrences. Since a number cannot be both even and strange in this situation, it is evident that the events are mutually exclusive. Therefore, P(A U B) would be 3/6 + 3/6 = 1 since a conventional die only produces odd and even numbers.

The alternative case, in which occurrences A and B are not mutually exclusive, is computed using the calculator above. In this instance

**P(A U B) is equal to P(A) + P(B) - P(A B).**

Find the likelihood of rolling an even number or a number that is a multiple of three using the dice-rolling example. The six values of the dice are used to represent the set in this instance and are written as follows: S = {1,2,3,4,5,6}

**Odds of getting an even number:**

P(A) = {2,4,6} = 3/6

**Probability of a 3+ number:**

P(B) = {3,6} = 2/6

**The point where A and B meet:**

P(A ∩ B) = {6} = 1/6

P(A U B) = 3/6 + 2/6 -1/6 = 2/3

the only OR of A and B

The calculator above can also calculate P(A XOR B), as depicted in the Venn diagram below. The event that A or B happens, but not both, is known as the "Exclusive OR" operation. The calculation looks like this:

Consider, for illustration, that two candy buckets, one containing Snickers and the other carrying Reese's, are placed outside the house on Halloween. The buckets of candy are surrounded by numerous neon flashing signs warning trick-or-treaters to only take one Snickers OR one Reese's, but not both! However, it is unlikely that every child will abide by the flashing neon signs. Calculate the likelihood that Snickers or Reese's will be chosen, but not both, given that the probability of a child exercising restraint while taking into account the negative effects of a potential future cavity is P(unlikely) = 0.001 and P(A) = 0.65 for Reese's and P(B) = 0.349 for Snickers, respectively.

0.65 + 0.349 - 2 × 0.65 × 0.349 = 0.999 - 0.4537 = 0.5453

Snickers or Reese's, but not both, have a 54.53 per cent probability of being picked.