Sure & Impossible Events: Probability Values

In probability, events are classified based on their likelihood of occurring. A sure event is an event that is guaranteed to happen, and its probability is always 1. Conversely, an impossible event is an event that can never happen, and its probability is always 0.

Understanding sure and impossible events is foundational for grasping more complex probability concepts.

Formulae

• Probability of a Sure Event: $P(Sure \, Event) = 1$
• Probability of an Impossible Event: $P(Impossible \, Event) = 0$

Examples

Example-1: Rolling a standard six-sided die.

• Sure Event: The event “rolling a number less than 7” is a sure event. $P(\text{Roll} < 7) = 1$ because every possible outcome (1, 2, 3, 4, 5, or 6) satisfies the condition.
• Impossible Event: The event “rolling a number greater than 6” is an impossible event. $P(\text{Roll} > 6) = 0$ because there are no numbers greater than 6 on a standard die.

Example-2: Drawing a card from a standard deck of 52 playing cards.

• Sure Event: The event “drawing a card that is either red or black” is a sure event. Since all cards are either red or black, this event is certain. $P(\text{Red or Black}) = 1$.
• Impossible Event: The event “drawing a card that is a unicorn” is an impossible event. There are no unicorn cards in a standard deck. $P(\text{Unicorn Card}) = 0$.

Theorem with Proof

Theorem: The probability of any event E, denoted as $P(E)$, always lies between 0 and 1 (inclusive): $0 \le P(E) \le 1$. This implies that if $P(E) = 1$, E is a sure event, and if $P(E) = 0$, E is an impossible event.

Proof:

Let S be the sample space (the set of all possible outcomes) of an experiment. An event E is a subset of S. The probability of an event is defined as:

$P(E) = \frac{\text{Number of favorable outcomes in E}}{\text{Total number of possible outcomes in S}}$

1. **For the lower bound:** The number of favorable outcomes in E cannot be negative. The smallest possible number of favorable outcomes is 0 (when E is an impossible event). Therefore, $P(E) \ge 0$.

2. **For the upper bound:** The number of favorable outcomes in E can be at most equal to the total number of possible outcomes in S (when E is a sure event). Therefore, the number of favorable outcomes in E is always less than or equal to the total number of possible outcomes in S. This means $P(E) \le 1$.

Combining the above, we get $0 \le P(E) \le 1$. If $P(E) = 1$, E includes all possible outcomes (it’s a sure event). If $P(E) = 0$, E contains no possible outcomes (it’s an impossible event).

Common mistakes by students

• Confusing terms: Students sometimes mix up “sure” with “likely”. A sure event is *guaranteed*, not just likely.
• Misunderstanding the boundaries: Thinking probabilities can be less than 0 or greater than 1.
• Incorrectly identifying events: Difficulty in determining whether an event will certainly happen or never happen. For example, students might mistakenly categorize an event with a very high probability (like “the sun will rise tomorrow”) as a sure event.

Real Life Application

Quality Control: In manufacturing, ensuring that a product meets certain specifications represents a ‘sure event’ if the quality control processes are perfectly implemented (which is ideal). Detecting a defect within a completely flawless batch of product would be an impossible event (under the same ideal circumstance). In reality, no event is a ‘sure event’ but probability helps determine that how often a ‘sure event’ or ‘impossible event’ can be possible.

Fun Fact

The concept of probability, and consequently, sure and impossible events, has its roots in games of chance! Mathematicians like Blaise Pascal and Pierre de Fermat, while studying games, developed early theories of probability that laid the groundwork for its application across many fields.

Recommended YouTube Videos for Deeper Understanding

Q.1 What is the probability of rolling a standard six-sided die and getting a number less than 7?

Check Solution

Ans: C

A standard die has numbers 1 through 6. All of these are less than 7. Therefore, the event is certain.

Q.2 A bag contains only red marbles. What is the probability of selecting a blue marble from the bag?

Check Solution

Ans: B

The event is impossible as there are no blue marbles in the bag.

Q.3 The probability of an event is 1. Which of the following statements is true?

Check Solution

Ans: C

A probability of 1 means the event will definitely happen.

Q.4 A spinner has four equal sections, colored red, blue, green, and yellow. What is the probability of the spinner landing on either red, blue, green, or yellow?

Check Solution

Ans: D

The spinner must land on one of the colors listed, meaning the event is certain.

Q.5 What is the probability of the sun rising in the east tomorrow?

Check Solution

Ans: C

The sun will certainly rise in the east.

Next Topic: Range of Probability