Conditional probabilities in a context

"Cows can become afflicted with a non-infectious disease irrespective of whether or not other cows are infected. A farmer has a herd of 200 cows. Seventy per cent of the herd have been treated by an injection that combats the disease but does not prevent its occurrence. The chances of recovery from the disease in one week are 1 in 20 if untreated and 1 in 5 if treated."

Question posed: "If a cow has recovered within one week, what is the probability that the cow received the injection?"

I am competent with statistics, yet I am confused which values to use in this instance for the given question. I understand I will use conditional probability.

$\left[P(R|I)=\frac{1}{5}\right]$- this is what I have concluded so far, are these correct (I=injection, R=recovered)

$\left[P(R|I^{\circ})=\frac{1}{20}\right]$-I have used the degree as I bar.

Rather than giving a straight answer I leave something for you to think about.

First write down formally all the probabilities that the exercise provides for example $$P(Recovered \mid Untreated)=1/20.$$

Second look at the two basic theorems that will get you trough almost all exercises like this: Law of total probability and Bayes theorem. The part one is to help you use these theorems.

I hope this will get you started.

• I was just typing that, what I have so far, are conditional probabilities. Would I be correct in saying: Given the cow has received an injection the probability it will recover in a week is 1/5? Commented Feb 26, 2018 at 17:54
• Correct. $P(Recovered \mid Treated)=1/5$ Commented Feb 26, 2018 at 17:57
• So would I use the conditional probability equation to obtain P(R ∩ I), Is this what my first question is asking? Commented Feb 26, 2018 at 17:59
• Or am I infact being asked, P(I|R) ? Commented Feb 26, 2018 at 18:08
• The last comment is what you are asked. That is $P(I \mid R)$. Commented Feb 26, 2018 at 19:09