Probability of an Impossible Event

The probability of an Impossible event is 0. When exploring the probability of an impossible event, we are looking at scenarios that simply cannot occur. You might ask yourself, what is an impossible event? In the context of statistics, this is an event that contains no outcomes from the sample space. Effectively, it is an empty set. Because there are no favorable outcomes to count, the ratio of favorable outcomes to total possible outcomes is zero.

Understanding the impossible event in probability is fundamental to mastering the scale of probability, which ranges from 0 to 1. The probability of impossible event represents the absolute bottom of this scale, indicating total certainty that the event will not happen. So, if you are asked “what is the probability of an impossible event?”, you can confidently state that it is zero. An impossible event is commonly represented as:

\(P(\text{Impossible Event}) = 0\)

Example: A deck of cards

To illustrate this, consider a standard deck of 52 playing cards. We know that in a standard deck, Hearts and Diamonds are red, while Clubs and Spades are black.

• The Experiment: Drawing a single card from the deck.
• The Event: Drawing a "Red Spade".

Since all Spades are black by definition, a "Red Spade" does not exist in the sample space. Therefore, drawing one is an impossible event.

\(P(\text{Red Spade}) = \frac{\text{Number of Red Spades}}{\text{Total Cards}} = \frac{0}{52} = 0\)
 

To find the probability of any event, divide the number of favorable outcomes by the total number of possible outcomes. When there is an event with an outcome that is impossible to include in the sample space, we can say that this is an impossible event and the probability of it happening is 0 

Example: Selecting a number

Consider a bag containing 5 balls, numbered 1 through 5.

• The experiment: Picking one ball from the bag at random.
• The event: Picking a ball numbered 10.

The sample space (all possible outcomes) is \(\{1, 2, 3, 4, 5\}\). Since the number 10 is not inside the bag, there are no favorable outcomes for this event.

\(P(\text{Picking a 10}) = \frac{\text{Count of balls numbered 10}}{\text{Total number of balls}} = \frac{0}{5} = 0\)

Does the probability of an impossible event mean empty set?

Yes, the two concepts are directly linked. In set theory notation, the sample space (S) represents everything that can happen. An impossible event is represented by the empty set (denoted as \(\emptyset\) or \(\{\}\)), because it contains no elements from that sample space.

Since the empty set has a size (cardinality) of 0, the probability calculation becomes 0 divided by the total sample size, which always results in 0.

\(P(\emptyset) = 0\)
 

Understanding impossible events helps clarify the concept of zero probability in real-life situations. These tips guide you to identify and differentiate such events effectively.

• Remember that the probability of an impossible event is always zero.
   
• Understand the difference between impossible events and rare events with very low probability.
   
• Use real-world examples to clearly identify situations that can never occur.
   
• Practice problems to distinguish impossible events from complementary and certain events.
   
• Visualize impossible events using probability trees or sample spaces for better clarity.
   

• Fill a bag with only green blocks and ask the student to pull out a red one. When they physically can't, explain that because there are zero red blocks, the probability is naturally zero.
   

• Use silly examples like "a dog speaking French" or "rolling a 7 on a die." The humor helps them distinguish between things that are just "hard" and things that are truly "impossible."
   

• Draw a probability line from 0 to 1. Mark the 0 end as "Impossible" using a bright color. This gives them a visual anchor showing that impossible events are at the very bottom of the scale.
   

• Clarify the difference! Winning the lottery is unlikely (a tiny chance), but picking a square marble from a bag of round ones is impossible (zero chance).
  Gamify the Empty Set: Ask them to list "Days of the week ending in Z." When they stare at a blank page, connect that empty list directly to the mathematical number 0.
   

• Hand them a deck of cards and challenge them to find a "Red Spade." Searching for something that breaks the rules reinforces why the count is zero.
   

Determining the probability of impossible events is used in research or manufacturing. Here are a few real-world applications:


Cybersecurity: The probability of an impossible event applies when blocking unauthorized access due to incorrect credentials. 

Traffic regulation: Some vehicles only start if the seatbelt is fastened. The probability of driving in these cars without the seatbelt fastened is 0.

Lotteries: Most lottery systems are programmed to accept valid number ranges. If a person selects a number that is outside the number range, the probability of winning is 0.


Quality control: In manufacturing, the probability of producing a completely defect-free product forever is considered an impossible event, helping set realistic quality goals.


Weather prediction: Predicting zero rainfall in a desert region during a heavy monsoon season is treated as an impossible event, showing the limits of certain weather outcomes.