Steps to Find the Likelihood of Occurrence of Events in Probability

Probability is a measure of the likelihood that a particular event will occur. It is expressed as a number between 0 and 1, where 0 indicates that the event will not happen and 1 indicates that the event will certainly happen. To find the probability of an event, follow these steps:

1. Determine the Sample Space
2. Determine the Number of Favorable Outcomes
3. Calculate the Probability

Let's break down each step with detailed explanations and examples.

Step 1: Determine the Sample Space

The sample space is the set of all possible outcomes of an experiment. Identifying the sample space is the first and fundamental step in calculating probability.

Example 1: Rolling a Die

• Experiment: Rolling a six-sided die.
• Sample Space (S): $\{1, 2, 3, 4, 5, 6\}$

The sample space includes all six possible outcomes of rolling a die.

Example 2: Drawing a Card from a Deck

• Experiment: Drawing a card from a standard deck of 52 cards.
• Sample Space (S): All 52 cards in the deck, such as {Ace of Hearts, 2 of Hearts, ..., King of Spades}.

The sample space includes all 52 possible cards that can be drawn from the deck.

Step 2: Determine the Number of Favorable Outcomes

A favorable outcome is an outcome that satisfies the event for which we are calculating the probability. Identify the number of favorable outcomes within the sample space.

Example 1: Rolling a Die

• Event: Rolling an even number.
• Favorable Outcomes (F): $\{2, 4, 6\}$
• Number of Favorable Outcomes: 3

In this case, there are three outcomes that satisfy the event of rolling an even number.

Example 2: Drawing a Card from a Deck

• Event: Drawing a King.
• Favorable Outcomes (F): $\{\text{King of Hearts, King of Diamonds, King of Clubs, King of Spades}\}$
• Number of Favorable Outcomes: 4

Here, there are four outcomes that satisfy the event of drawing a King.

Step 3: Calculate the Probability

To calculate the probability of an event, divide the number of favorable outcomes by the total number of possible outcomes (the size of the sample space).

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

Example 1: Rolling a Die

• Total Number of Possible Outcomes (Sample Space): 6
• Number of Favorable Outcomes: 3

$P(\text{Rolling an even number}) = \frac{3}{6} = 0.5$

The probability of rolling an even number on a six-sided die is 0.5.

Example 2: Drawing a Card from a Deck

• Total Number of Possible Outcomes (Sample Space): 52
• Number of Favorable Outcomes: 4

$P(\text{Drawing a King}) = \frac{4}{52} = \frac{1}{13} \approx 0.0769$

The probability of drawing a King from a standard deck of 52 cards is approximately 0.0769.

Additional Examples

Example 3: Tossing a Coin

• Experiment: Tossing a fair coin.
• Sample Space (S): $\{\text{Heads, Tails}\}$
• Event: Getting Heads.
• Favorable Outcomes: $\{\text{Heads}\}$
• Number of Favorable Outcomes: 1
• Total Number of Possible Outcomes: 2

$P(\text{Getting Heads}) = \frac{1}{2} = 0.5$

The probability of getting Heads when tossing a fair coin is 0.5.

Example 4: Picking a Red Ball

• Experiment: Picking a ball from a bag containing 3 red balls and 2 blue balls.
• Sample Space (S): $\{\text{Red, Red, Red, Blue, Blue}\}$
• Event: Picking a red ball.
• Favorable Outcomes: $\{\text{Red, Red, Red}\}$
• Number of Favorable Outcomes: 3
• Total Number of Possible Outcomes: 5

$P(\text{Picking a Red Ball}) = \frac{3}{5} = 0.6$

The probability of picking a red ball from the bag is 0.6.

Conclusion

Calculating the probability of an event involves determining the sample space, identifying the number of favorable outcomes, and dividing the number of favorable outcomes by the total number of possible outcomes. By following these steps, you can accurately determine the likelihood of various events occurring in different probabilistic scenarios. Understanding and applying these steps is fundamental to mastering probability theory and its applications.