Three coins are tossed simultaneously. Consider the event E three heads or three tails, F at least two heads and G at most two heads. Of the pairs (E, F), (E, G) and (F, G), which are independent? which are dependent?
✓
Answer
The sample space of the experiment is given by
S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
Clearly, E = {HHH, TTT}, F = {HHH, HHT, HTH, THH}
and G = {HHT, HTH, THH, HTT, THT, TTH, TTT}
Also E $\cap$ F = {HHH}, E $\cap$ G = {TTT}, F $\cap$ G = { HHT, HTH, THH}
Therefore P(E) = $\frac{2}{8}=\frac{1}{4}$, P(F) = $\frac{4}{8}=\frac{1}{2}$, P(G) = $\frac{7}{8}$
and $\mathrm{P}(\mathrm{E} \cap \mathrm{F})=\frac{1}{8}$, $\mathrm{P}(\mathrm{E} \cap \mathrm{G})=\frac{1}{8}$, $\mathrm{P}(\mathrm{F} \cap \mathrm{G})=\frac{3}{8}$
Also, P(E).P(F) = $\frac{1}{4} \times \frac{1}{2}=\frac{1}{8}$, P(E).P(G) = $\frac{1}{4} \times \frac{7}{8}=\frac{7}{32}$ and P(F).P(G) = $\frac{1}{2} \times \frac{7}{8}=\frac{7}{16}$
Thus P(E $\cap$ F) = P(E) . P(F)
$\mathrm{P}(\mathrm{E} \cap \mathrm{G}) \neq \mathrm{P}(\mathrm{E}) \cdot \mathrm{P}(\mathrm{G})$
and $\mathrm{P}(\mathrm{F} \cap \mathrm{G}) \neq \mathrm{P}(\mathrm{F}) \cdot \mathrm{P}(\mathrm{G})$
Hence, the events (E and F) are independent, and the events (E and G) and (F and G) are dependent.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.