Probability of Impossible Events: Explanation, Applications , and Solved Examples

The probability of an impossible event is always zero, since E = 0 for an impossible event, hence P(E) = 0.

• Probability is a measure of the possibility or chance that an event will occur.
• It is a number ranging from 0 to 1, with 0 indicating an impossible event and 1 indicating a certain event.
• To determine the probability of an event, divide the number of favourable outcomes by the total number of possible outcomes.
• In probability theory, an impossible event is one that cannot occur under any circumstances, and hence it has a probability of zero.
• It is the opposite of a sure event, which is guaranteed to occur with probability one.

Key Terms: Probability, Sample space, Events, Types of events, Null set, Empty set, Rolling a dice, Tossing a coin, Compound Event, Simple Event

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What are Events in Probability?

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A probability event is a set of outcomes from experiments.

• In other words, an event in probability is a subset of the sample space.
• The sample space, also known as the individual space of a random experiment, is the entire set of possible outcomes.
• Probability refers to the possibility that an event will occur.
• The probability of any event occurring is between 0 and 1.

The sample space for tossing three coins concurrently is given by

S = {(T , T , T) , (T , T , H) , (T , H , T) , (T , H , H ) , (H , T , T ) , (H , T , H) , (H , H, T) ,(H , H , H)}

Suppose we wish to find just the outcomes that have at least two heads; the set of all such possibilities can be given as

E = { (H , T , H) , (H , H ,T) , (H , H ,H) , (T , H , H)}

Hence, an event is a subset of the sample space; for example, E is a subset of S.

Types of Events in Probability

The following are types of events that can occur in probability.

• Simple Event
• Compound Event 
• Sure or Certain Event
• Impossible Event
• Equivalent Event or Identical Event
• Equally Likely Event
• Exhaustive Event
• Favourable Event
• Mutually Exclusive Event

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Probability of Impossible Events

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The probability of an impossible event refers to the possibility of an event not occurring at all and having no chance of happening in the future. Thus, the chance of such an event is always equal to zero.

Probability describes the possibility of an event occurring in numerical terms.

• We experience several real-life situations in which we guess the result of an occurrence.
• We may be certain or uncertain about the outcome of an event.
• A sure event, as the term implies, is one that is certain to occur.
• For example, when flipping a coin, the result will always be either head or tail, which is known as the sure event.
• Thus, the probability of a certain happening is one, whereas the probability of an impossible event is zero.

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Impossible Events and Empty Sets

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An impossible event is one that has no chance of occurring. It is important to distinguish between impossible and unlikely events, which have a non-zero probability of occurrence but a low chance.

• An empty set, often called a null set, is a set that has no elements.
• In probability theory, an empty set can be used to denote an impossible event.
• Consider rolling a 7 on an ordinary six-sided die.
• This impossible event can be represented as the set {7}, which is an empty set since no element in the set corresponds to the outcome of 7.
• Similarly, the empty set {} represents the event of flipping a coin and landing on both heads and tails at the same time.
• It is important to highlight that the probability of an empty set is always 0 because no possible outcomes exist.
• In other words, the probability of an impossible event is always 0, which can be represented by the empty set in probability theory.

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Finding the Probability of Impossible Event

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The probability of an impossible event is zero. By definition, an impossible event is one that cannot occur and so has a probability of 0.

• To further understand this concept, keep in mind that the probability of each event must fall between 0 and 1.
• A probability of zero indicates that the occurrence is impossible, but a probability of one indicates that the event is certain to occur.

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Applications of Probability in Daily Life

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Probability plays an important role in our daily lives, and we meet it in a variety of ways, including:

Gambling

Probability is the foundation of gambling. Games of chance, such as roulette and slot machines, are dependent on the probability of specific outcomes.

Risk Assessment

Probability is used in risk assessment to determine the possibility of an event occurring and its possible consequences. This is essential in areas like insurance, finance, and safety planning.

Medical Diagnosis

Probability is used in medical diagnosis to assess if a patient has a certain disease based on symptoms and test findings.

Weather Forecasts

Weather forecasts are based on probability models that use data from previous weather patterns to estimate future conditions.

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Solved Examples of the Probability of Impossible Events

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Ques. Find the probability of drawing a card with the number 9 out of 26 English alphabet cards.

Ans. Given that there are 26 English alphabet cards, therefore

Total number of outcomes = 26

We have to find the probability of getting the number 9.

Getting a card of number 9 is an impossible event in English alphabet cards.

So, the number of favourable outcomes = 0.

⇒ P(E) = 0.

As a result, the probability of the impossible event, i.e. drawing a card with the number 9 out of 26 alphabet cards, is equal to zero.

Ques. What is the probability of getting five aces from a deck of 52 cards?

Ans. Given a deck of 52 cards, therefore

Number of outcomes = 52

We have to find the probability of getting five aces from a deck of 52 cards.

Since a deck of 52 cards only has 4 aces, Hence

The number of favourable outcomes = 0.

⇒ P(E) = 0.

Therefore, the probability of the impossible event, i.e., drawing five aces from a pack of 52 cards, is equal to zero.

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Things To Remember

• Probability is a measure of the possibility or chance that an event will occur.
• The probability of any event occurring is between 0 and 1.
• An impossible event is one that cannot occur under any circumstances.
• The sample space is the entire set of possible outcomes.
• An empty set, often called a null set, is a set that has no elements.
• The probability of a certain event is 1, whereas the probability of an impossible event is 0.