Probability of two events occurring together independent

The probability of an event is the chance that the event will occur in a given situation. The probability of getting "tails" on a single toss of a coin, for example, is 50 percent, although in statistics such a probability value would normally be written in decimal format as 0.50. The individual probability values of multiple events can be combined to determine the probability of a specific sequence of events occurring. To do so, however, you must know if the events are independent or not.

First, watch the video below for a quick refresher on basic probability:

1. Determine the individual probability (P) of each event that is to be combined. Calculate the ratio m/M where m is the number of outcomes that result in the event of interest and M is all possible outcomes. For example, the probability of rolling a six on a single die roll can be calculated using m = 1 (since only one face gives a result of six) and M = 6 (since there are six possible faces that could turn up) for P = 1/6 or 0.167.
2. Determine if the two individual events are independent or not. Independent events are not influenced by each other. The probability of heads on a coin toss, for instance, is not affected by the results of a prior toss of the same coin and so is independent.
3. Determine if the events are independent. If not, adjust the probability of the second event to reflect the conditions specified for the first event. For example, if there are three buttons -- one green, one yellow, one red -- you may wish to find the probability of picking the red and then the green button. P for picking the first button red is 1/3 but P for picking the second button green is 1/2 since one button is now gone.
4. Multiply the individual probabilities of the two events together to obtain the combined probability. In the button example, the combined probability of picking the red button first and the green button second is P = (1/3)(1/2) = 1/6 or 0.167.
   

Tip: This same approach can be used to find the probability of more than two events.

Related Articles

How to Use Scientific Calculators to Do Probability

How to Find the Probability of Two Spinners

How to Calculate Weighted Probabilities

How to Use a Calculator to Figure Percentage

How to Use a Binomial Table

How to Explain the Sum & Product Rules of Probability

How to Find Genotype Ratio

How to Calculate the Percent of Something

How to Calculate Binomial Probability

To Calculate Arcsine, What Buttons Do You Press on...

How to Use Scientific Calculators to Do Probability

How to Find the Z Score on a TI-83

How to Do a Punnett Square

How to Calculate Probability

How to Calculate Probability and Normal Distribution

How to Calculate the Interquartile Range

How to Calculate Dice Probabilities

This Is Why It's So Tough to Get a Perfect March Madness...

References

• University of Texas at Austin: Combining Probabilities: R. Fitzpatrick

About the Author

Michael Judge has been writing for over a decade and has been published in "The Globe and Mail" (Canada's national newspaper) and the U.K. magazine "New Scientist." He holds a Master of Science from the University of Waterloo. Michael has worked for an aerospace firm where he was in charge of rocket propellant formulation and is now a college instructor.

Probability is a type of ratio where we compare how many times an outcome can occur compared to all possible outcomes.

$$Probability=\frac{The\, number\, of\, wanted \, outcomes}{The\, number \,of\, possible\, outcomes}$$

Example

What is the probability to get a 6 when you roll a die?

A die has 6 sides, 1 side contain the number 6 that give us 1 wanted outcome in 6 possible outcomes.

Independent events: Two events are independent when the outcome of the first event does not influence the outcome of the second event.

When we determine the probability of two independent events we multiply the probability of the first event by the probability of the second event.

$$P(X \, and \, Y)=P(X)\cdot P(Y)$$

To find the probability of an independent event we are using this rule:

Example

If one has three dice what is the probability of getting three 4s?

The probability of getting a 4 on one die is 1/6

The probability of getting 3 4s is:

$$P\left ( 4\, and\, 4\, and\, 4 \right )=\frac{1}{6}\cdot \frac{1}{6}\cdot\frac{1}{6}=\frac{1}{216}$$

When the outcome affects the second outcome, which is what we called dependent events.

Dependent events: Two events are dependent when the outcome of the first event influences the outcome of the second event. The probability of two dependent events is the product of the probability of X and the probability of Y AFTER X occurs.

$$P(X \, and \, Y)=P(X)\cdot P(Y\: after\: x)$$

Example

What is the probability for you to choose two red cards in a deck of cards?

A deck of cards has 26 black and 26 red cards. The probability of choosing a red card randomly is:

$$P\left ( red \right )=\frac{26}{52}=\frac{1}{2}$$

The probability of choosing a second red card from the deck is now:

$$P\left ( red \right )=\frac{25}{51}$$

The probability:

$$P\left ( 2\,red \right )=\frac{1}{2}\cdot \frac{25}{51}=\frac{25}{102}$$

Two events are mutually exclusive when two events cannot happen at the same time. The probability that one of the mutually exclusive events occur is the sum of their individual probabilities.

$$P(X \, or \, Y)=P(X)+ P(Y)$$

An example of two mutually exclusive events is a wheel of fortune. Let's say you win a bar of chocolate if you end up in a red or a pink field.

What is the probability that the wheel stops at red or pink?

P(red or pink)=P(red)+P(pink)

$$P\left (red \right )=\frac{2}{8}=\frac{1}{4}$$

$$P\left (pink \right )=\frac{1}{8}$$

$$P\left ( red\, or\, pink \right )=\frac{1}{8}+\frac{2}{8}=\frac{3}{8}$$

Inclusive events are events that can happen at the same time. To find the probability of an inclusive event we first add the probabilities of the individual events and then subtract the probability of the two events happening at the same time.

$$P\left (X \, or \, Y \right )=P\left (X \right )+ P\left (Y \right )-P\left (X \, and \, Y \right )$$

Example

What is the probability of drawing a black card or a ten in a deck of cards?

There are 4 tens in a deck of cards P(10) = 4/52

There are 26 black cards P(black) = 26/52

There are 2 black tens P(black and 10) = 2/52

$$P\left ( black\, or\, ten \right )=\frac{4}{52}+\frac{26}{52}-\frac{2}{52}=\frac{30}{52}-\frac{2}{52}=\frac{28}{52}=\frac{7}{13}$$

What is the probability of 2 independent events happening together?

Can two independent events occur together?

How do you combine probabilities for independent events?

How do you find the probability of two independent variables?