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

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.