91Ó°ÊÓ

10,000 tickets are sold in one lottery, and 5000 tickets are sold in another lottery.

A person owns 100 tickets in each lottery.

Let:

\({W_1} = \)The event that she will win first prize in the first lottery

\({W_2} = \) The event that she will win first prize in the second lottery

The Additional rule of probability provides the probability of occurrence of either of the events A or B. Mathematically; it is given by:

\({\bf{P}}\left( {{\bf{A}} \cup {\bf{B}}} \right){\bf{ = P}}\left( {\bf{A}} \right){\bf{ + P}}\left( {\bf{B}} \right) - {\bf{P}}\left( {{\bf{A}} \cap {\bf{B}}} \right)\)

Since the events of winning first prize in first and second lotteries are independent; therefore,

\(P\left( {{W_1} \cap {W_2}} \right) = P\left( {{W_1}} \right) \times P\left( {{W_2}} \right)\)

So, using the Additional Rule of Probability, obtaining the probability that she will win at least one first prize as:

\(P\left( {{\rm{at least one first prize}}} \right) = P\left( {{W_1} \cup {W_2}} \right)\)

That is,

\(\begin{aligned}{}P\left( {{W_1} \cup {W_2}} \right) &= P\left( {{W_1}} \right) + P\left( {{W_2}} \right) - P\left( {{W_1} \cap {W_2}} \right)\\ &= \frac{{100}}{{10000}} + \frac{{100}}{{5000}} - P\left( {{W_1}} \right) \times P\left( {{W_2}} \right)\\ &= 0.01 + 0.02 - 0.01 \times 0.02\\ &= 0.0298\end{aligned}\)

Therefore, the required probability is 0.0298.