Introduction to Dependent Events in Probability

Mar 12, 2021, 5:42:10 AM Tech and Science

Probability is the likelihood of occurrence of an event. It ranges from 0 to 1 where 0 indicates impossibility and 1 certainty. An experiment or trial is an infinitely repeated process and which has a well-defined set of outcomes termed as sample space. The possible result of an experiment is called outcome. For example, in the experiment of tossing a coin, the outcome is either heads or tails. The sample space is {H, T}. An event is a set of outcomes of an experiment in which it is assigned with probability. The different types of events in probability are as follows:

●       Simple event

●       Compound event

●       Certain event

●       Impossible event

●       Equally likely event

●       Mutually exclusive events

●       Complementary events

●       Independent and dependent events

●       Exhaustive events

●       Identical events

●       Favorable events

Dependent Events

To know dependent events, the concept of independent events is essential.

An independent event is defined as an event that doesn’t have any impact on the occurrence of another subsequent event in an experiment. The experiment of rolling dice can be an example of an independent event. Here the sample space is {1, 2, 3, 4, 5, and 6}.

If the occurrence of one event is affected by the occurrence of another event, then the events are said to be dependent on nature. For instance, you are required to draw a colored ball from a bag of balls. If the first ball drawn is not replaced before the second draw, then the outcome of the second draw is affected by the first draw. This concept of dependent events in probability introduces the concept of conditional probability.

Conditional probability is defined as the probability of an event given that another event has already occurred. It is denoted as P (A | [given] B]. The formula for conditional probability is given by

P (A / B) = P (A ∩ B) / P(A) or P (B ∩ A) / P(A) given, P (A) > 0

For example, a card has to be drawn randomly from a deck of 52 cards. If the card drawn in the first instance isn’t replaced, then the card chosen in the second draw is out of 51 cards. So, the probability is being affected.

Hence, the probability of dependent events can be explained by the formula:

P (A and B) = P (A) * P (B|A)

Solved Example on Dependent Events in Probability

1] A wallet contains 4 bills of 5 dollars, 5 bills of 10 dollars and 3 bills of 20 dollars. 2 bills are chosen randomly without replacement. Find the P (drawing a 5 dollar bill followed by a 5 dollar bill).

Solution:

Number of 5 dollar bills = 4

Total number of bills = 12

P (drawing a 5 dollar bill) = 4 / 12

The probability of the second draw affected the first.

Number of 5 dollar bills left = 3

A total of 11 bills are left.

P (drawing a 5 dollar bill after a 5 dollar bill) = 3 / 11

P (drawing a 5 dollar bill followed by a 5 dollar bill) = P (drawing a 5 dollar bill) * P (drawing a 5 dollar bill after a 5 dollar bill) = $\frac{4}{12}*\frac{3}{11}=\frac{1}{11}$

Hence the P (drawing a 5 dollar bill followed by a 5 dollar bill) = 1 / 11

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