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# In order to gauge the effectiveness of the OHaganBooks.com site, you recently commissioned a survey of online shoppers. According to the results, 2% of online shoppers visited OHaganBooks.com during a one-week period, while 5% of them visited at least one of OHaganBooks.com’s two main competitors: JungleBooks.com and FarmerBooks.com. Use this information to answer. Assuming that visiting OHaganBooks.com was independent of visiting a competitor, what percentage of online shoppers visited either OHaganBooks.com or a competitor?

Solution

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Denote the events as follows:

\begin{align*} & H = \qty{\text{a randomly chosen online shopper visited OHaganBooks.com}}\\ & C = \qty{\text{a randomly chosen online shopper visited either of the main competitors}} \end{align*}

We are given the following information:

\begin{align*} & P(H) = 2\% = 0.02\\ & P(C) = 5\% = 0.05 \end{align*}

Also, $H$ and $C$ are said to be independent which means that $P(H\cap C) = P(H)P(C)$.

We want to find the percentage of online shoppers that visited either OHaganBooks.com or a competitor i.e. we want to find $P(H\cup C)$. Using the addition principle for probabilities, we get

\begin{align*} P(H\cup C) &= P(H)+P(C)-P(H\cap C)\\ &= P(H)+P(C)-P(H)P(C)\\ &= 0.02+0.05-0.02\times0.05\\ &= 0.069\\ &=6.9\% \end{align*}

Therefore, 6.9% of online shoppers visited either OHaganBooks.com or a competitor.

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