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Sure Event

Sure Event

An event that has a 100% probability of occurring.

 

 

See also

Impossible event

Key Formula

P(S)=1P(S) = 1
Where:
  • P(S)P(S) = The probability of the sure event S
  • 11 = Certainty — the maximum possible probability

Worked Example

Problem: A standard die has faces numbered 1 through 6. What is the probability that you roll a number less than 7?
Step 1: Identify the sample space. When you roll a standard die, the possible outcomes are 1, 2, 3, 4, 5, and 6.
Sample space Ω={1,2,3,4,5,6}\text{Sample space } \Omega = \{1, 2, 3, 4, 5, 6\}
Step 2: Identify the favorable outcomes. Every face shows a number less than 7, so the event includes all six outcomes.
S={1,2,3,4,5,6}S = \{1, 2, 3, 4, 5, 6\}
Step 3: Calculate the probability. The number of favorable outcomes equals the total number of outcomes.
P(S)=66=1P(S) = \frac{6}{6} = 1
Answer: The probability of rolling a number less than 7 on a standard die is 1. This is a sure event because every possible outcome satisfies the condition.

Another Example

Problem: A bag contains 4 red marbles, 3 blue marbles, and 5 green marbles. What is the probability of drawing a marble that is red, blue, or green?
Step 1: Count the total number of marbles in the bag.
4+3+5=124 + 3 + 5 = 12
Step 2: Count the favorable outcomes. Since every marble in the bag is either red, blue, or green, all 12 marbles satisfy the condition.
Favorable outcomes=12\text{Favorable outcomes} = 12
Step 3: Compute the probability.
P(red or blue or green)=1212=1P(\text{red or blue or green}) = \frac{12}{12} = 1
Answer: The probability is 1. Drawing a marble that is red, blue, or green is a sure event because those are the only colors available.

Frequently Asked Questions

What is the difference between a sure event and an impossible event?
A sure event has a probability of 1, meaning it always happens. An impossible event has a probability of 0, meaning it can never happen. They are exact opposites on the probability scale.
Is the sample space itself always a sure event?
Yes. The sample space contains every possible outcome of an experiment, so at least one outcome in it must occur. This makes the sample space a sure event with probability 1. In formal notation, P(Ω) = 1.

Sure Event vs. Impossible Event

A sure event (probability = 1) is the opposite of an impossible event (probability = 0). A sure event includes every outcome in the sample space, so it must occur. An impossible event contains no outcomes from the sample space, so it can never occur. Every other event falls between these two extremes, with a probability strictly between 0 and 1.

Why It Matters

The sure event anchors the top end of the probability scale. All probability axioms depend on the fact that the entire sample space has probability 1, which is precisely the statement that the sample space is a sure event. Recognizing sure events also helps you quickly simplify problems — if every outcome satisfies a condition, there is no calculation needed beyond noting the probability is 1.

Common Mistakes

Mistake: Thinking that a very likely event (e.g., probability 0.99) is a sure event.
Correction: A sure event must have a probability of exactly 1. An event with probability 0.99 is highly likely but can still fail to occur. Only when every single outcome in the sample space satisfies the condition is the event truly sure.
Mistake: Confusing the sure event with a certain outcome.
Correction: A sure event is an event (a set of outcomes) that is guaranteed to occur, not a single predetermined result. For example, rolling a number from 1 to 6 on a die is a sure event, but you still don't know which specific number will appear.

Related Terms