Table of Content

• What are Events in Probability?
• What are Dependent Events in Probability?
• Dependent Events Definition
• Dependent Events Examples
• Probability of Dependent Events
• Dependent Events Formula
• Difference Between Independent and Dependent Events
• Dependent Events: Examples in Real Life
• Dependent Events – Solved Examples
• Practice Problems on Dependent Events

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.

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• Probability in Maths
• Conditional Probability
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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?

Dependent Events – FAQs

What are dependent events in probability?

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.

How do you identify if two events are dependent?

To determine if two events are dependent, assess whether the occurrence of the first event changes the probability of the second event. If the probability changes, the events are dependent​.

Can you provide an example of dependent events?

A classic example is drawing cards from a deck without replacement. If a card is not returned to the deck, it affects the probabilities of drawing subsequent cards because the composition of the deck has changed​.

What impact does the first event have on a subsequent dependent event?

In dependent events, the first event alters the probability space for the subsequent event. For instance, if a red ball is drawn from a bag without replacement, the likelihood of drawing another red ball changes because the total number of red balls and the total number of balls in the bag are both reduced​.

How does replacing an item affect whether events are dependent or independent?

Replacement is key to determining dependence. If an item is replaced after an event, the events are independent because the initial conditions are restored for each event. Without replacement, the conditions change, making the events dependent​.