Theoretical probability

Theoretical Probability

Theoretical probability is the likelihood of an event occurring based on the possible outcomes in a sample space. It is calculated using a formula that relates the number of favourable outcomes to the total number of outcomes.

Definitions

1. **Sample Space**: The sample space (denoted as S) is the set of all possible outcomes of a probability experiment. For example, when tossing a fair six-sided die, the sample space is S = {1, 2, 3, 4, 5, 6}.

2. **Event**: An event is a specific outcome or a set of outcomes from the sample space. For instance, rolling an even number (2, 4, or 6) is an event.

Calculating Theoretical Probability

The theoretical probability (P) of an event A is given by the formula:

P(A) = Number of favourable outcomes / Total number of outcomes

Let’s look at an example to understand this better.

Example 1: Tossing a Die

Suppose you want to find the probability of rolling a 3 on a fair six-sided die.

• **Favourable outcomes**: There is only one favourable outcome, which is rolling a 3.
• **Total outcomes**: The total number of outcomes is 6 (the numbers 1 to 6).

Using the formula:

P(rolling a 3) = Number of favourable outcomes / Total number of outcomes = 1 / 6

Thus, the theoretical probability of rolling a 3 is 1/6.

Example 2: Drawing Cards

Consider a standard deck of 52 playing cards. What is the probability of drawing an Ace?

• **Favourable outcomes**: There are 4 Aces in a deck (Ace of hearts, diamonds, clubs, and spades).
• **Total outcomes**: The total number of cards is 52.

Using the formula:

P(drawing an Ace) = Number of favourable outcomes / Total number of outcomes = 4 / 52 = 1 / 13

So, the theoretical probability of drawing an Ace is 1/13.

Complementary Events

The complement of an event A (denoted as A') is the event that A does not occur. The probability of the complement can be calculated as:

P(A') = 1 - P(A)

For example, if the probability of rolling a 3 is 1/6, then the probability of not rolling a 3 is:

P(not rolling a 3) = 1 - P(rolling a 3) = 1 - 1/6 = 5/6

Example 3: Probability of Not Rolling a Specific Number

Using the die example again, let’s find the probability of not rolling a 2.

• **Favourable outcomes**: The favourable outcomes for not rolling a 2 are {1, 3, 4, 5, 6}, which gives us 5 outcomes.
• **Total outcomes**: The total number of outcomes remains 6.

Using the formula:

P(not rolling a 2) = 5 / 6

Dependent and Independent Events

Events can either be dependent or independent:

• **Independent Events**: Two events A and B are independent if the occurrence of one does not affect the occurrence of the other. For example, tossing a coin and rolling a die.
• **Dependent Events**: Two events A and B are dependent if the occurrence of one event affects the occurrence of the other. For example, drawing two cards from a deck without replacement.

Example 4: Independent Events

Let’s consider the probability of flipping a coin (getting heads) and rolling a die (getting a 4).

• **Probability of heads**: P(H) = 1/2
• **Probability of rolling a 4**: P(4) = 1/6

Since these events are independent, the combined probability is:

P(H and 4) = P(H) × P(4) = (1/2) × (1/6) = 1/12

Example 5: Dependent Events

Now, consider drawing two cards from a standard deck without replacement. What is the probability of drawing an Ace followed by a King?

• **Probability of drawing an Ace first**: P(Ace) = 4/52
• **Probability of drawing a King second (after drawing an Ace)**: P(King | Ace) = 4/51 (since there are still 4 Kings but only 51 cards left).

Using the multiplication rule for dependent events:

P(Ace and King) = P(Ace) × P(King | Ace) = (4/52) × (4/51) = 16/2652 = 4/663

Summary

• Theoretical probability is calculated as the number of favourable outcomes divided by the total number of outcomes.
• The complement of an event A is the probability that A does not occur.
• Events can be independent or dependent, affecting how their probabilities are calculated.

Check your understanding

1. What is the theoretical probability of rolling a 5 on a fair six-sided die?
2. In a standard deck of cards, what is the probability of drawing a Queen?
3. If the probability of an event A is 3/10, what is the probability of its complement A'?
4. When drawing two cards from a deck without replacement, what is the probability of drawing a heart followed by a diamond?