Question

a. how many total outcomes are possible?
b. p(x) = \frac{4}{10}
c. p(y) = \frac{6}{10}
d. p(x\cap y) = \frac{1}{10}
e. p(x|y) =
f. are events x and y independent events? why or why not?

Explanation:

Step1: Calculate total outcomes

Add all the numbers in the Venn - diagram: \(3 + 1+5 + 1=10\).

Step2: Calculate \(P(X)\)

Number of elements in \(X\) is \(3 + 1=4\). So \(P(X)=\frac{4}{10}\).

Step3: Calculate \(P(Y)\)

Number of elements in \(Y\) is \(1 + 5=6\). So \(P(Y)=\frac{6}{10}\).

Step4: Calculate \(P(X\cap Y)\)

Number of elements in \(X\cap Y\) is \(1\). So \(P(X\cap Y)=\frac{1}{10}\).

Step5: Calculate \(P(X|Y)\)

Use the formula \(P(X|Y)=\frac{P(X\cap Y)}{P(Y)}\). Substitute \(P(X\cap Y)=\frac{1}{10}\) and \(P(Y)=\frac{6}{10}\), we get \(P(X|Y)=\frac{\frac{1}{10}}{\frac{6}{10}}=\frac{1}{6}\).

Step6: Check independence

Two events \(X\) and \(Y\) are independent if \(P(X\cap Y)=P(X)\times P(Y)\).
\(P(X)\times P(Y)=\frac{4}{10}\times\frac{6}{10}=\frac{24}{100}=\frac{6}{25}
eq\frac{1}{10}\).

Answer:

a. \(10\)
b. \(\frac{4}{10}\)
c. \(\frac{6}{10}\)
d. \(\frac{1}{10}\)
e. \(\frac{1}{6}\)
f. Events \(X\) and \(Y\) are not independent. Because \(P(X\cap Y)
eq P(X)\times P(Y)\)