# Sample space of the following situation

$$10\%$$ of the people that goes in holiday go to the beach. The others go to the desert. The probability to get a sunburn at the beach is $$\frac{3}{10}$$ whereas to get a sunburn at the desert is $$\frac{7}{10}$$.

1) What is the sample space of the situation ?

2) What is the probability that a person that went for holiday got a sunburn ?

3) Given that a person comeback without a sunburn, what is the probability that this person where at the beach ?

My problem is only for 1). If $$\Omega$$ is the sample of event, then the sample event is the $$\sigma -$$algebra generated by the events $$B$$: "go to the beach" and $$S$$: "got a sunburn". But what can be the sample space ? Is it $$\Omega =\{B\cap S, B^c\cap S, B\cap S^c, B^c\cap S^c\} \ \ ?$$

But my problem is that $$B$$ ans $$S$$ are event, so it should be something as $$B=\{\omega \in \Omega \mid \omega \text{ go to the beach}\}$$, so $$\Omega$$ should be a set of people, no ? Or maybe $$\Omega =\{(i,j)\mid i\in \{B,B^c\},\{S,S^c\}\},$$ should be more correct ?

What do you think ?

## 1 Answer

You can choose your sample space as you like. You could take $$\Omega=\{B \cap S,B^c \cap S, B \cap S^c, B^c \cap S^c \}$$.

But this events have not the same probability e.g. $$P(B \cap S)=0.1 \cdot \frac{3}{10}$$ and $$P(B^c \cap S^c)=0.9\cdot \frac{3}{10}$$.

You could also choose $$\Omega$$ as the set of all persons (let's say there are $$n$$ persons). You have $$|\Omega|=n$$ and $$P(\omega)=\frac{1}{n}$$. Further $$B$$ is the subset ($$B \subset \Omega$$) with all people that go to the beach and $$P(B)=\frac{|B|}{|\Omega|}$$.

The questions 2) and 3) are missing some informations e.g. the probability that somebody goes to the beach during the winter.

If the questions 3) is: "3) Given that a person comeback without a sunburn, what is the probability that this person was on the beach" you can answer this questions with Bayes law: $$P(\text{Beach}|\text{sunburn})=\frac{0.1\cdot \frac{3}{10}}{0.1\cdot \frac{3}{10}+0.9\cdot \frac{7}{10}}\approx 0.045$$

• In the case I choose the second way (i.e. choosing $n$ people), what will be $\mathbb P\{\omega \}$ ? (because such a measure should be extendable to $\mathcal P(\Omega )$ and not only on $\sigma \{S,B\}$. – user601023 Oct 23 '18 at 17:13
• $P(\omega)=\frac{1}{n}$ – garondal Oct 23 '18 at 17:26