Question

a. how many total outcomes are possible? 10
b. p(k) = 5/10
c. p(l) = 2/10
d. p(k∩l) =
e. p(k|l) =
f. are events k and l independent events? why or why not?

Explanation:

Step1: Calculate total outcomes

Count all the numbers in the diagram. There are \(4 + 1+1 + 4=10\) total outcomes.

Step2: Calculate \(P(K)\)

Event \(K\) has \(4 + 1=5\) favorable outcomes. So \(P(K)=\frac{5}{10}\)

Step3: Calculate \(P(L)\)

Event \(L\) has \(1+1 = 2\) favorable outcomes. So \(P(L)=\frac{2}{10}\)

Step4: Calculate \(P(K\cap L)\)

The intersection \(K\cap L\) has \(1\) favorable outcome. So \(P(K\cap L)=\frac{1}{10}\)

Step5: Calculate \(P(K|L)\)

Use the formula \(P(K|L)=\frac{P(K\cap L)}{P(L)}\). Substitute \(P(K\cap L)=\frac{1}{10}\) and \(P(L)=\frac{2}{10}\). Then \(P(K|L)=\frac{\frac{1}{10}}{\frac{2}{10}}=\frac{1}{2}\)

Step6: Check for independence

Two events \(K\) and \(L\) are independent if \(P(K\cap L)=P(K)\times P(L)\)
\(P(K)\times P(L)=\frac{5}{10}\times\frac{2}{10}=\frac{10}{100}=\frac{1}{10}\) and \(P(K\cap L)=\frac{1}{10}\)

Answer:

a. \(10\)
b. \(\frac{5}{10}\)
c. \(\frac{2}{10}\)
d. \(\frac{1}{10}\)
e. \(\frac{1}{2}\)
f. Yes, because \(P(K\cap L)=P(K)\times P(L)=\frac{1}{10}\)