Discussion
A coin is tossed thrice. Let \(X\) be the event that head occurs in each of the first two tosses. Let \(Y\) be the event that a tail occurs on the third toss. Let \(Z\) be the event that two tails occur in three tosses. Based on the above information, which one of the following statements is TRUE?
Answer: B
The true statement is Y and Z are dependent.
There are 8 equally likely outcomes. \(X = \{HHH, HHT\}\), \(Y\) = tail on the third toss (4 outcomes), \(Z\) = exactly two tails = \(\{HTT, THT, TTH\}\).
\[ \begin{aligned} P(Y)P(Z) &= \tfrac{1}{2} \cdot \tfrac{3}{8} = \tfrac{3}{16} \\ P(Y \cap Z) &= P(\{HTT, THT\}) = \tfrac{2}{8} = \tfrac{4}{16} \end{aligned} \]
These differ, so Y and Z are dependent, which also rules out 'Y and Z are independent'. For X and Y, \(P(X \cap Y) = P(HHT) = \frac{1}{8} = \frac{1}{4} \cdot \frac{1}{2}\), so they are independent and 'X and Y are not independent' is false. X and Z cannot both happen, while each has positive probability, so they are dependent.
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