Theoretical Probability

The concept of probability is a fundamental aspect of mathematics that deals with the likelihood of an event occurring. Among the different types of probability, theoretical probability plays a vital role as it provides a framework to predict the outcomes of different situations based on logical reasoning, without any actual experimentation. Understanding theoretical probability helps a person to figure out the possibilities in a given scenario based on the known information.

Understanding theoretical probability

Theoretical probability is defined as the ratio of the number of favorable outcomes to the number of possible outcomes, provided that each outcome is equally likely. This provides an ideal way to calculate probabilities without the need for experimental data.

Mathematically, theoretical probability can be expressed as:

Theoretical Probability, P(E) = Number of Favorable Outcomes / Total Number of Possible Outcomes

Where:

• Favorable outcomes: outcomes that meet the criteria for the event we are interested in.
• Possible Outcomes: All possible outcomes in a given scenario.

Example 1: Tossing a fair coin

One of the simplest examples of theoretical probability is tossing a fair coin. A normal coin has two sides: heads and tails. When we talk about a fair coin, it means that both sides, heads and tails, are equally likely to occur when tossed.

Let us calculate the probability of getting heads when tossing a coin:

• Total Possible Outcomes: When a coin is tossed, there are two possible outcomes – heads or tails.
• Favorable outcome: The event we are interested in is getting heads.

Number of favorable outcomes = 1 (heads) Total possible outcomes = 2 (heads, tails) Theoretical Probability, P(getting heads) = 1 / 2 = 0.5

Thus, the theoretical probability of getting heads when tossing a fair coin is 0.5 or 50%.

Example 2: Throwing fair dice

Dice are another common example when discussing probability. A standard dice has six faces, numbered 1 to 6. When the dice is thrown, each face has an equal chance of coming up face up.

Let's calculate the probability of getting 4:

• Total possible outcomes: When a dice is thrown, there are six possible outcomes from 1 to 6.
• Favourable Outcome: The event we are interested in is that a 4 comes up.

Number of favorable outcomes = 1 (rolling a 4) Total possible outcomes = 6 (1, 2, 3, 4, 5, 6) Theoretical Probability, P(rolling a 4) = 1 / 6 ≈ 0.1667

Thus, the theoretical probability of a fair die rolling a 4 is approximately 0.1667, or about 16.67%.

Example 3: Drawing a red card from the deck

Consider a standard deck of 52 cards. These cards are divided into four suits: hearts, diamonds, spades and clubs. Hearts and diamonds are red, while spades and clubs are black.

Let's calculate the probability of drawing a red card from the deck:

• Total possible outcomes: There are 52 cards in total.
• Favorable outcome: The event we are interested in is drawing a red card. There are 26 red cards: 13 hearts and 13 diamonds.

Number of favorable outcomes = 26 (13 hearts + 13 diamonds) Total possible outcomes = 52 Theoretical Probability, P(drawing a red card) = 26 / 52 = 1 / 2 = 0.5

Therefore, the theoretical probability of drawing a red card from a deck of 52 cards is 0.5 or 50%.

Theoretical probability with multiple events

In scenarios with multiple events, theoretical probability can also help determine the probability of each event occurring independently or simultaneously. Let us see how theoretical probability applies in such scenarios.

Example 4: Probability of throwing more than one dice

Suppose two dice are being thrown simultaneously. Each dice is fair and has six sides numbered 1 to 6. We are interested in finding the probability of getting a sum of 8 from the rolls of the dice.

When two dice are thrown, each dice has 6 possible outcomes, making up the total combinations of outcomes:

Total possible outcomes = 6 (for first die) × 6 (for second die) = 36

Next, determine the favorable outcomes that add up to 8:

• (2, 6)
• (3, 5)
• (4, 4)
• (5, 3)
• (6, 2)

There are a total of 5 favorable outcomes.

Number of favorable outcomes = 5 Theoretical Probability, P(sum of 8) = 5 / 36 ≈ 0.1389

Therefore, the theoretical probability of getting a sum of 8 from two fair dice is approximately 0.1389, or 13.89%.

Properties of theoretical probability

There are several basic properties of theoretical probability that make it a useful concept in understanding and solving probability-related problems:

• Range: The probability of any event lies between 0 and 1. A probability of 0 means the event will not happen, while a probability of 1 indicates certainty.
• Total Probability: The sum of the probabilities of all possible outcomes of a trial is always 1.
• Complementary events: The probability of an event not occurring is obtained by subtracting 1 from the probability of it occurring:

  P(not E) = 1 - P(E)

Example 5: Complementary Events

Returning to the example of tossing a coin, let us determine the probability of not getting heads (i.e. getting tails).

• Total possible outcomes: 2 (heads, tails)
• Favorable outcome for tails: tails

Theoretical Probability, P(tails) = 1 / 2 = 0.5

Alternatively, using the complementary probability:

P(tails) = 1 - P(heads) = 1 - 0.5 = 0.5

Both approaches confirm that the probability of getting tails is either 0.5 or 50%.

Conclusion

Theoretical probability provides a logical and systematic way to determine the likelihood of possible outcomes in various scenarios. By considering each possible outcome and determining which outcomes are most favorable to the event being examined, individuals can easily figure out the associated probabilities.

Whether tossing a coin, throwing a dice or drawing a card, the principles of theoretical probability remain constant. Understanding these principles is the basis for understanding more advanced concepts in probability, statistics and decision-making processes in various disciplines.

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