Understanding the Probability Problem

The question asks us to find the probability that neither event E nor event F occurs. This is represented as P(not E and not F). We are given the individual probabilities of events E and F, P(E) and P(F), and the probability that both E and F occur, P(E and F).

We are given the following probabilities:

• P(E) = $\frac{5}{8}$
• P(F) = $\frac{1}{2}$
• P(E and F) = P(E $\cap$ F) = $\frac{1}{4}$

We need to calculate P(not E and not F), which can be written as P(E' $\cap$ F'), where E' is the complement of E (not E) and F' is the complement of F (not F).

Applying Probability Rules: De Morgan's Law

To find P(not E and not F), we can use De Morgan's Law in probability, which states that the complement of the union of two events is the intersection of their complements. Mathematically:

$(E \cup F)' = E' \cap F'$

Therefore, P(E' $\cap$ F') = P((E $\cup$ F)').

Using the complement rule, P(A') = 1 - P(A), we can write:

P((E $\cup$ F)') = 1 - P(E $\cup$ F)

So, P(not E and not F) = 1 - P(E $\cup$ F).

Now, our task is to find P(E $\cup$ F), the probability that event E or event F or both occur.

Calculating the Probability of the Union of Events E and F

The formula for the probability of the union of two events E and F is:

P(E $\cup$ F) = P(E) + P(F) - P(E $\cap$ F)

We have all the values needed to use this formula:

• P(E) = $\frac{5}{8}$
• P(F) = $\frac{1}{2}$
• P(E $\cap$ F) = $\frac{1}{4}$

Let's substitute these values into the formula:

P(E $\cup$ F) = $\frac{5}{8} + \frac{1}{2} - \frac{1}{4}$

To add and subtract these fractions, we need a common denominator. The least common multiple of 8, 2, and 4 is 8. We convert the fractions:

• $\frac{1}{2} = \frac{1 \times 4}{2 \times 4} = \frac{4}{8}$
• $\frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}$

Now, substitute the equivalent fractions back into the formula:

P(E $\cup$ F) = $\frac{5}{8} + \frac{4}{8} - \frac{2}{8}$

Perform the addition and subtraction:

P(E $\cup$ F) = $\frac{5 + 4 - 2}{8}$

P(E $\cup$ F) = $\frac{9 - 2}{8}$

P(E $\cup$ F) = $\frac{7}{8}$

So, the probability that event E or F or both occur is $\frac{7}{8}$.

Finding P(not E and not F)

Now that we have P(E $\cup$ F), we can find P(not E and not F) using the relationship we derived earlier:

P(not E and not F) = 1 - P(E $\cup$ F)

Substitute the calculated value of P(E $\cup$ F):

P(not E and not F) = $1 - \frac{7}{8}$

To subtract the fraction from 1, write 1 as a fraction with the same denominator as the other fraction, which is 8:

$1 = \frac{8}{8}$

So, the calculation becomes:

P(not E and not F) = $\frac{8}{8} - \frac{7}{8}$

P(not E and not F) = $\frac{8 - 7}{8}$

P(not E and not F) = $\frac{1}{8}$

Summary of Calculation Steps

The probability P(not E and not F) is $\frac{1}{8}$.

Revision Table: Key Probability Concepts

Additional Information: Visualizing Probability with Venn Diagrams

Venn diagrams can be helpful in visualizing probability concepts like union and intersection. For two events E and F:

• The sample space (total possible outcomes) is the rectangle.
• Event E is represented by a circle, and its area corresponds to P(E).
• Event F is represented by another circle, and its area corresponds to P(F).
• The overlapping region of the circles is the intersection, E $\cap$ F, representing P(E $\cap$ F).
• The combined area of both circles is the union, E $\cup$ F, representing P(E $\cup$ F). This is the sum of the areas minus the overlap (which is counted twice).
• The area outside both circles represents the outcomes where neither E nor F occurs. This is (E $\cup$ F)', which is also E' $\cap$ F'. The probability of this region is 1 - P(E $\cup$ F).

In this problem, finding P(not E and not F) is finding the probability of the region outside both circles in the Venn diagram. We calculated the probability of the combined circles (E $\cup$ F) and subtracted it from the total probability (1).