Suppose $A, B, C$ are events in a common probability space with $P(A) = 0.2, \ P(B) = 0.2, \ P(C) = 0.3$, $P(A \cap B) = 0.1, \ P(A \cap C) = 0.1, \ P(B \cap C) = 0.1$. Which of the following are possible values of $P(A \cup B \cup C)$?
To determine the possible values of \(P(A \cup B \cup C)\), we can use the principle of inclusion-exclusion in probability. This formula is given by:
\(P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(C \cap A) + P(A \cap B \cap C)\)
We know:
- \(P(A) = 0.2\)
- \(P(B) = 0.2\)
- \(P(C) = 0.3\)
- \(P(A \cap B) = 0.1\)
- \(P(A \cap C) = 0.1\)
- \(P(B \cap C) = 0.1\)
Let's compute with the assumption that \(P(A \cap B \cap C) = 0\), as we are not given its value:
\(\begin{align*} P(A \cup B \cup C) &= 0.2 + 0.2 + 0.3 - 0.1 - 0.1 - 0.1 + 0 \\ &= 0.6 - 0.3 \\ &= 0.3 \end{align*}\)
This result shows that if \(P(A \cap B \cap C) = 0\), then \(P(A \cup B \cup C) = 0.3\).
However, the problem doesn't provide \(P(A \cap B \cap C)\), so we need to consider possible values to find other potential values of \(P(A \cup B \cup C)\). Let's check potential maximum overlap when \(P(A \cap B \cap C) = 0.1\).
\(\begin{align*} P(A \cup B \cup C) &= 0.2 + 0.2 + 0.3 - 0.1 - 0.1 - 0.1 + 0.1 \\ &= 0.6 - 0.2 \\ &= 0.4 \end{align*}\)
If we further assume \(P(A \cap B \cap C) = 0.2\), let's calculate:
\(\begin{align*} P(A \cup B \cup C) &= 0.2 + 0.2 + 0.3 - 0.1 - 0.1 - 0.1 + 0.2 \\ &= 0.6 - 0.1 \\ &= 0.5 \end{align*}\)
Therefore, the possible values of \(P(A \cup B \cup C)\) consistent with the provided information are \(0.4\) and \(0.5\).
Thus, the correct options are:
0.5
0.4
Important Questions from Discrete Probability
- A biased six-faced die is tossed once. Suppose that the probability of any prime number showing up is twice that of any non-prime number showing up. Then, the probability that an odd number will show up is
- Let $X$ and $Y$ be independent Poisson random variables with means $4$ and $2$, respectively. Then, which of the following statements are true?
- Consider the M/M/1 queue in which customers arrive according to a Poisson process with rate $3$ and successive service times are independent exponential random variables having mean $\frac{1}{9}$. Let $P_n$ be the long run probability that there are exactly $n$ customers in the system. Then, which of the following statements are true?
- Let $X$ be a Binomial$(n, p)$ random variable, where $n \in \{5,6\}$ and $p\in \{\frac{1}{4}, \frac{3}{4}\}$. If $X = 3$ is observed, then the maximum likelihood estimate of $(n, p)$ is
- Suppose two fair dice are thrown independently at random. Let $X$ and $Y$ be the numbers on the upper face of the first die and that of the second die, respectively. Then which of the following statements are true?
