Dependent Events

Dependent Events in Probability: Dependent events are those events that are affected by the outcomes of events that had already occurred previously. i.e. Two or more events that depend on one another are known as dependent events. If one event is by chance changed, then another is likely to differ.

In this article, we will discuss Dependent Events in detail, including their examples, theorem, as well as the method to calculate the probability of dependent events, and the difference between Dependent Events and Independent Events.

What are Events in Probability?

In probability theory, an event is a specific outcome or a set of outcomes of an experiment or a random phenomenon.

Events can range from simple outcomes, such as flipping a coin and getting heads, to more complex outcomes involving multiple trials or conditions. Events can be of many types such as

• Simple Events
• Compound Events
• Mutually Exclusive Events
• Dependent Events
• Independent Events

Read in Detail: Events in Probability – Types, Examples, Definition

Dependent Events in Probability?

Dependent events in probability are those that rely on previous outcomes, where the occurrence of one event affects the probability of another event happening.

For example, if you draw a red card on the first draw, the probability of drawing another red card on the second draw will change because there are now fewer red cards in the deck.

Dependent Events Definition

Dependent events in probability are events whose occurrence of one affects the probability of occurrence of the other.

Dependent Events Examples

Some examples os dependent events include:

• Consider a bag containing red and blue marbles. If you draw a red marble from the bag without replacement, the probability of drawing another red marble decreases because there are now fewer red marbles in the bag.

• Suppose you’re predicting weather patterns. The occurrence of rain on one day might increase the likelihood of rain on the following day due to atmospheric conditions.

Read More: Types of Events in Probability

Probability of Dependent Events

Probability of dependent events can be calculated using conditional probability. Conditional probability is the probability of an event occurring given that another event has already occurred.

For dependent events A and B, the probability of event B occurring given that event A has already occurred is denoted as P(B∣A).

Note: Formula for conditional probability is given by P(A∩B) = P(A) × P(B∣A).

Dependent Events Formula

Formula for probability of dependent events is given as:

P(B∣A) = P(A∩B) / P(A)

Where,

• P(A∩B) represents the probability of both events A and B occurring.
• P(A) is the probability of event A occurring.
• P(B∣A) is the conditional probability of event B occurring given that event A has already occurred.

Dependent Events and Independent Events

The key distinction between independent and dependent events lies in how the outcome of one event affects the probability of another. In the following table all the major differences between both dependent and independent events are given:

Aspect	Dependent Events	Independent Events
Definition	Events where the outcome of one event affects the outcome of another.	Events where the outcome of one event does not affect the outcome of another.
Dependency	The occurrence of one event influences the probability of the other event.	The occurrence of one event has no influence on the probability of the other event.
Sampling	Often occurs when sampling without replacement.	Typically occurs when sampling with replacement.
Conditional Probability	Conditional probability differs from marginal probability.	Conditional probability equals marginal probability.
Example	Drawing cards from a deck without replacement.	Flipping a coin and rolling a die simultaneously.
Formula	P(A∩B) = P(A) × P(B∣A)	P(A∩B) = P(A) × P(B)

Read More: Dependent and Independent Events in Probability

Dependent Events: Examples in Real Life

• Preparing a Meal: Let’s say you are assembling a sandwich. Whether or not you remembered to buy bread at the grocery earlier will determine whether it is available. A sandwich without bread demonstrates how one occurrence depends on another.
• Weather Affects Traffic: The weather has a big impact on how much traffic there is when you commute to work. Perhaps you could get through your commute quickly on a sunny day. If there is snow, however, delays should be expected. Here, the weather has a direct impact on the traffic situation.
• Seeing a Movie Sequel: Whether or not you’ve seen the original film (Event A) can have a significant impact on how well you comprehend and appreciate a sequel (Event B). You can miss important narrative points if you don’t watch the original.
• Planting a Garden: Whether or not you correctly prepared the soil and planted the seeds (Event A) will determine the outcome of your garden (Event B). You wouldn’t have a garden if you never planted any seeds. There is a direct link between the two occurrences.
• Academic Performance: Attendance at classes and completion of assignments during the semester are major factors in determining one’s performance on a final test. Ignoring homework or skipping classes can cause you to grasp the content poorly, which can negatively impact your exam score.

Summary – Dependent Events

In conclusion, dependent events occur when the outcome of one event influences the outcome of another. This dependency often arises when sampling without replacement or in situations where the sample space changes based on previous outcomes. In this article, we have discussed dependent events in detail.

People Also Read:

• Probability in Maths
• Conditional Probability
• Card Probability
• Coin Toss Probability

Dependent Events – Solved Examples

Question 1: You have a bag containing 3 red marbles and 2 blue marbles. If you draw one marble note its color and then draw another marble without replacement, what is the probability of drawing two red marbles in a row?

Solution:

The probability of drawing a red marble on the first draw is 3/5

If a red marble is drawn on the first draw, there are now 2 red marbles and 4 marbles in total.

So, the probability of drawing another red marble on the second draw is 2/4

Therefore, the probability of drawing two red marbles in a row is 3/5 × 1/2 = 3/10

Question 2: You have a standard deck of 52 cards. If you draw one card, note its suit and then draw another card without replacement, what is the probability of drawing two hearts in a row?

Solution:

The probability of drawing a heart on the first draw is 13/52 = 1/4

If a heart is drawn on the first draw, there are now 12 hearts and 51 cards left in the deck. So, the probability of drawing another heart on the second draw is 12/51

Therefore, the probability of drawing two hearts in a row is :

= 1/4 × 12/51 = 4/17

Question 3: You flip two fair coins sequentially. What is the probability of getting two heads in a row?

Solution:

Since the events are dependent, the probability of getting two heads in a row is the product of the probability of getting a head on the first flip (which is 1/2) and the probability of getting a head on the second flip (also 1/2) which equals :

= 1/2 × 1/2 = 1/4

Question 4: You have a deck of 52 cards. If you draw one card, note its value (2 through 10) and then draw another card without replacement, what is the probability of drawing two face cards in a row?

Solution:

There are 12 face cards in a deck (4 kings, 4 queens, and 4 jacks). The probability of drawing a face card on the first draw is= 12/52

= 12/52 = 3/13

If a face card is drawn on the first draw, there are now 11 face cards and 51 cards left in the deck. So, the probability of drawing another face card on the second draw is = 11/51

Therefore, the probability of drawing two face cards in a row is:

3/13 × 11/51

 = 33/221

Question 5: You have 5 pairs of socks in a drawer each pair consisting of one black sock and one white sock. If you randomly draw two socks from the drawer without replacement, what is the probability of drawing two black socks?

Solution:

The probability of drawing a black sock on the first draw is: 5/10 = 1/2

If a black sock is drawn on the first draw, there are now 4 black socks and 9 socks left in the drawer. So, the probability of drawing another black sock on the second draw is = 4/9

Therefore, the probability of drawing two black socks in a row is:

= 1/2 × 4/9

= 2/9

Practice Problems on Dependent Events

Problem 1: In a basket, there are 4 apples, 3 oranges and 2 bananas. If you randomly select two fruits from the basket without replacement, what is the probability of selecting two oranges in a row?

Problem 2: You have a standard deck of 52 cards. If you draw one card, note its value (2 through 10), and then draw another card without replacement, what is the probability of drawing two consecutive cards with values adding up to 11?

Problem 3: You have a bag containing 4 red marbles and 6 blue marbles. If you draw one marble, note its color, and then draw another marble without replacement, what is the probability of drawing two blue marbles in a row?

Problem 4: You have 5 pairs of socks in a drawer each pair consisting of one blue sock and one black sock. If you randomly draw two socks from the drawer without replacement, what is the probability of drawing two blue socks?

Problem 5: You flip two fair coins sequentially. What is the probability of getting two tails in a row?

FAQs on Dependent Events

What are Dependent Events?

Dependent events are events in probability theory where the outcome of one event affects the outcome of another event.

How do dependent events differ from independent events?

In dependent events, the outcome of one event influences the probability of another event whereas in independent events, the outcome of one event does not affect the probability of another event.

What is an Example of Dependent Event in Everyday Life?

One example of dependent events is drawing marbles from a bag without replacement. If you draw a red marble from the bag, the probability of drawing another red marble decreases because there is now one less red marble in the bag.

How do you calculate the probability of dependent events?

To calculate the probability of dependent events, we use the formula for conditional probability.

Are all events in probability dependent?

No, not all events are dependent. Independent events, where the outcome of one event does not affect the outcome of another, also exist.