Probability of Union of Independent Events

We are given three mutually independent events E, F, and G with their probabilities:

• $P(E) = 0.4$
• $P(F) = 0.6$
• $P(G) = 0.8$

We need to find the probability $P(\bar{E} \cup \bar{F} \cup G)$.

Calculating Complement Probabilities

First, calculate the probabilities of the complement events:

• $P(\bar{E}) = 1 - P(E) = 1 - 0.4 = 0.6$
• $P(\bar{F}) = 1 - P(F) = 1 - 0.6 = 0.4$
• $P(\bar{G}) = 1 - P(G) = 1 - 0.8 = 0.2$

Strategy: Using the Complement Rule

It's easier to calculate the probability of the complement of the desired event and subtract it from 1. The complement of $(\bar{E} \cup \bar{F} \cup G)$ is $\overline{(\bar{E} \cup \bar{F} \cup G)}$.

Using De Morgan's laws, we have:

$\overline{(\bar{E} \cup \bar{F} \cup G)} = \overline{\bar{E}} \cap \overline{\bar{F}} \cap \bar{G} = E \cap F \cap \bar{G}$

Calculating Intersection Probability

Since E, F, and G are mutually independent, the events E, F, and $\bar{G}$ are also mutually independent.

Therefore, the probability of their intersection is the product of their individual probabilities:

$P(E \cap F \cap \bar{G}) = P(E) \times P(F) \times P(\bar{G})$

$P(E \cap F \cap \bar{G}) = 0.4 \times 0.6 \times 0.2$

$P(E \cap F \cap \bar{G}) = 0.24 \times 0.2 = 0.048$

Final Probability Calculation

Now, we can find the desired probability using the complement rule:

$P(\bar{E} \cup \bar{F} \cup G) = 1 - P(E \cap F \cap \bar{G})$

$P(\bar{E} \cup \bar{F} \cup G) = 1 - 0.048$

$P(\bar{E} \cup \bar{F} \cup G) = 0.952$