Mastering Conditional Probability and Limits: Math Activities

Activity 10: Conditional Probability

Objective

To explain the computation of conditional probability of a given event, assuming event B has already occurred, using the example of throwing a pair of dice.

Method of Construction

• Paste a white paper on a piece of plywood of a convenient size.
• Create a square and divide it into 36 unit squares (1cm each).
• Write the pairs of numbers representing the outcomes.

Demonstration

1. The figure displays all possible outcomes of the experiment, representing the sample space.
2. Suppose we find the conditional probability of event A given event B has occurred, where A is "a number 4 appears on both dice" and B is "4 appears at least once." We must find P(A|B).
3. From the figure: Number of outcomes favorable to B = 11; Number of outcomes favorable to A∩B = 1.
4. P(B) = 11/36; P(A∩B) = 1/36; P(A|B) = P(A∩B)/P(B) = 1/11.

Observations

• n(A) =
• n(B) =
• n(A∩B) =
• P(A∩B) =
• P(A|B) =

Application

This activity is helpful in understanding the concept of conditional probability.

Activity 4: Limit of a Function

Objective

To find analytically the limit of a function f(x) at x=c and to check the continuity of the function at that point.

Method of Construction

1. Consider the function: f(x) = (x² - 16) / (x - 4).
2. Select points on the left and right sides of c (=4) that are very near to c.
3. Calculate the corresponding values of f(x) for each point.
4. Record the values of x and f(x) in a table.

Demonstration

The values of x and f(x) are recorded in a table to observe the behavior of the function as it approaches the limit.

Observation

1. The value of f(x) approaches ___ as x → 4 from the left.
2. The value of f(x) approaches ___ as x → 4 from the right.
3. Therefore, lim f(x) = ___ and lim f(x) = ___.
4. Thus, lim f(x) = ___, f(4) = ___.
5. Is lim f(x) = f(4)?
6. Since f(c) = lim f(x), the function is continuous at x=4.

Application

This activity is useful in understanding the concept of limits and the continuity of a function at a point.