How the Probability Calculation Works

Probability is a measure of how likely an event is to occur. It is expressed as a number between 0 and 1, where 0 means the event will not occur, and 1 means the event will certainly occur. To calculate the probability of an event, follow these steps:

1. Identify the event for which you want to calculate the probability.
2. Determine the total number of possible outcomes in the sample space (denoted as \( N_{\text{total}} \)).
3. Determine the number of favorable outcomes (denoted as \( N_{\text{favorable}} \)), or the outcomes that result in the event happening.
4. Use the probability formula to calculate the probability:
• Probability = \( \frac{N_{\text{favorable}}}{N_{\text{total}}} \)
5. Interpret the result. The probability is usually expressed as a decimal between 0 and 1, or as a percentage by multiplying the result by 100.

Probability is commonly used in various fields such as statistics, gambling, risk assessment, and decision-making to quantify uncertainty and predict the likelihood of events.

Extra Tip

Remember that the probability of an event is always between 0 and 1. If the probability is 0, the event will never happen. If the probability is 1, the event will certainly happen.

Example: Suppose you roll a fair six-sided die. What is the probability of rolling a 4?

• The total number of possible outcomes when rolling the die is 6 (since the die has 6 sides). So, \( N_{\text{total}} = 6 \).
• The number of favorable outcomes (rolling a 4) is 1, since there is only one "4" on the die. So, \( N_{\text{favorable}} = 1 \).
• Using the probability formula: \( \text{Probability} = \frac{1}{6} = 0.1667 \).

The probability of rolling a 4 is \( 0.1667 \), or about **16.67%**.

Complementary Probability

Complementary probability refers to the probability that an event will not occur. The complementary event is simply the opposite of the event you're interested in.

The formula for complementary probability is:

\[ P(\text{Not A}) = 1 - P(A) \]

For example, if the probability of an event occurring is 0.7, the probability of it not occurring is:

\[ P(\text{Not A}) = 1 - 0.7 = 0.3 \]

So, the probability of the event not occurring is **30%**.

Probability with Multiple Events

When calculating the probability of two or more events, there are different methods depending on whether the events are independent or dependent.

Independent Events

Two events are independent if the outcome of one does not affect the outcome of the other. For independent events, the probability of both events occurring is the product of their individual probabilities.

If \( A \) and \( B \) are independent events, the probability of both events occurring is:

\[ P(A \cap B) = P(A) \times P(B) \]

Example: Suppose you roll two fair dice. What is the probability of rolling a 4 on the first die and a 6 on the second die?

• The probability of rolling a 4 on the first die is \( \frac{1}{6} \).
• The probability of rolling a 6 on the second die is \( \frac{1}{6} \).
• The probability of both events occurring is: \( \frac{1}{6} \times \frac{1}{6} = \frac{1}{36} \), or about **2.78%**.

Dependent Events

Two events are dependent if the outcome of one event affects the outcome of the other. For dependent events, the probability of both events occurring is the product of the probability of the first event and the conditional probability of the second event given the first event.

If \( A \) and \( B \) are dependent events, the probability of both events occurring is:

\[ P(A \cap B) = P(A) \times P(B|A) \]

Example: If you draw two cards from a deck without replacement, what is the probability of drawing an Ace on the first draw and a King on the second draw?

• The probability of drawing an Ace on the first draw is \( \frac{4}{52} \) (since there are 4 Aces in a deck of 52 cards).
• After drawing an Ace, there are now 51 cards left, with 4 Kings still in the deck. The probability of drawing a King on the second draw is \( \frac{4}{51} \).
• The probability of both events occurring is: \( \frac{4}{52} \times \frac{4}{51} = \frac{16}{2652} \), or about **0.60%**.