Fair Die Roll Scenario Analysis
The problem describes a scenario where a standard fair die is rolled repeatedly until a face value that is not 5 or 6 appears. The variable X represents the value shown on this final roll.
Determining the Sample Space for X
The rolling stops when a face from the set $\{1, 2, 3, 4\}$ is observed. This means the possible values for the final roll, X, are restricted to $\{1, 2, 3, 4\}$.
Since the die is fair, the probability of rolling any specific face is $1/6$. The probability that the process stops on any given roll is the probability of rolling a 1, 2, 3, or 4, which is $P(\text{stop}) = P(\{1, 2, 3, 4\}) = 4/6 = 2/3$.
To find the probability distribution for X, we consider the conditional probability of each outcome given that the stopping condition is met (i.e., the roll is in $\{1, 2, 3, 4\}$):
$ P(X=i \mid X \in \{1, 2, 3, 4\}) = \frac{P(X=i)}{P(X \in \{1, 2, 3, 4\})} = \frac{1/6}{4/6} = \frac{1}{4} $Therefore, the effective sample space for X consists of the outcomes $\{1, 2, 3, 4\}$, and each outcome has an equal probability of $1/4$.
Analyzing Events A and B
Event A is defined as $\{X \text{ is even}\}$. Within the sample space $\{1, 2, 3, 4\}$, the outcomes satisfying event A are $\{2, 4\}$.
Event B is defined as $\{X \text{ is at most 2}\}$. Within the sample space $\{1, 2, 3, 4\}$, the outcomes satisfying event B are $\{1, 2\}$.
Calculating the Intersection Probability $P(A \cap B)$
The intersection event $A \cap B$ requires that X satisfies both condition A (being even) and condition B (being at most 2). The only outcome from the sample space $\{1, 2, 3, 4\}$ that meets both criteria is $X=2$.
Thus, the event $A \cap B$ is equivalent to the event $\{X=2\}$.
The probability of this intersection is:
$ P(A \cap B) = P(X=2) $Based on the conditional probability distribution established earlier:
$ P(A \cap B) = \frac{1}{4} $Important Questions from Probability (Notes)
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A canal system is shown in the figure.
Water flows from A to B through two channels. Gates $G_1$ and $G_2$ are operated independently to regulate the flow. Probability of $G_1$ to be open is 10% while that of $G_2$ is 20%. The probability that water will flow from A to B is
- Let $X$ and $Y$ be i.i.d. uniform $(0,1)$ random variables. Let $Z = \max(X,Y)$ and $W = \min(X,Y)$.
Then $P((Z-W) > 1/2)$ is
