Let S = {a, b, c, d, e} with P(a) = 0.13, P(b) = 0.1, P(c) = 0.19, and P(d) = 0.15. If F is the event {c, d, e}, find P(F). (Enter answer as a decimal with at least 2 correct decimal places)

We are given the set \( S = \{a, b, c, d, e\} \) with the probabilities of four elements: \( P(a) = 0.13 \), \( P(b) = 0.1 \), \( P(c) = 0.19 \), and \( P(d) = 0.15 \).

We are also given the event \( F = \{c, d, e\} \), and we are tasked with finding \( P(F) \).

The probability of event \( F \) is the sum of the probabilities of the elements in \( F \). This means:

\[
P(F) = P(c) + P(d) + P(e)
\]

However, the probability of \( e \) is not provided directly, but we can find it using the fact that the sum of all probabilities in a sample space must equal 1:

\[
P(a) + P(b) + P(c) + P(d) + P(e) = 1
\]

Substituting the known values:

\[
0.13 + 0.1 + 0.19 + 0.15 + P(e) = 1
\]

Solving for \( P(e) \):

\[
0.57 + P(e) = 1
\]

\[
P(e) = 1 - 0.57 = 0.43
\]

Now, we can compute \( P(F) \):

\[
P(F) = P(c) + P(d) + P(e) = 0.19 + 0.15 + 0.43 = 0.77
\]

Thus, the probability of event \( F \) is \( \boxed{0.77} \).

Would you like further details or have any questions?

Here are 5 related questions:
1. How would the probability change if \( P(c) \) were altered?
2. What happens to the total probability if one event's probability exceeds 1?
3. How do you calculate probabilities for disjoint or overlapping events?
4. How can we find conditional probabilities given this data?
5. How would we model this problem if the set \( S \) expanded to more than 5 elements?

**Tip:** Always ensure the sum of probabilities in any sample space equals 1; this is a key property of probability distributions.

Learn how to calculate the probability of event F = {c, d, e} in the sample space S = {a, b, c, d, e} where P(a) = 0.13, P(b) = 0.1, P(c) = 0.19, and P(d) = 0.15. Use the law of total probability to find the missing value of P(e) and solve for P(F). Suitable for Grades 9-12.

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