The Question
A company is considering a marketing campaign for a product which currently is purchased by $4\%$ of people. If the company decides to run a newspaper campaign then based on previous similar campaigns, the Market Research Company states that the probability that an arbitrary person (person selected at random) sees the advert is $0.55$.
Also, the Market Research Company assumes that a person who sees the advert will be twice as likely to purchase the product (i.e. $8\%$ of people). Given that a person had bought the product, what is the conditional probability that they had seen the adverts?
My Understanding
I'm just practicing for a basic maths exam and would just appreciate some clarity on how to calculate conditional probabilities. This is my attempt:
$$\frac{0.55 \cdot 0.08}{ 0.08} + \frac{0.45 \cdot 0.04}{0.04} = 0.2475$$
Am I heading in the right direction or am I completely wrong? Thanks!
Given the data
$\mathbb{P}[\text{Purchase}|\overline{\text{Adv}}]=0.04$
$\mathbb{P}[\text{Purchase}|\text{Adv}]=0.08$
You are requesting to calculate
$$\mathbb{P}[\text{Adv}|\text{Purchase}]=\frac{\mathbb{P}[\text{Adv}\cap\text{Purchase}]}{\mathbb{P}[\text{Purchase}]}=\frac{0.55\cdot0.08}{0.55\cdot0.08+0.45\cdot0.04}\approx 70.97\%$$