Workshop questions

Define what is meant by the following.

1. (a)

   The axioms of probability

2. (b)

   Conditional probability

3. (c)

   The law of total probability

4. (d)

   Bayes’ theorem

5. (e)

   Independence

Suppose that an experiment is performed $n$ times. For any event $A$ of the sample space let $n_A$ denote the number of times that the event $A$ occurs. Define $\mathrm{P}(A)=n_A/n$. Show that $\mathrm{P}(\cdot )$ satisfies the three axioms of probability.

Two dice are rolled $n$ times in succession.

1. (a)

   What is the probability that a double 6 appears at least once?

2. (b)

   How large does $n$ need to be to make this probability at least $\frac{1}{2}$?

Let $A$ and $B$ be events with probabilities $\mathrm{P}(A)=3/4$ and $\mathrm{P}(B)=1/3$. Use the addition rule to show that $\mathrm{P}(A\cap B)\ge 1/12$. Give an example of a sample space $\mathrm{\Omega }$, a probability $\mathrm{P}$, and events $A$ and $B$ in which $\mathrm{P}(A)=3/4$, $\mathrm{P}(B)=1/3$ and $\mathrm{P}(A\cap B)=1/12$.

Tomorrow there will be either rain or snow but not both. The probability of rain is $2/5$ and the probability of snow is $3/5$.

If it rains then the probability that I will be late for my lecture is $1/5$, while the corresponding probability in the event of snow is $3/5$.

1. (a)

   What is the probability that I will be late?

2. (b)

   If I am late, what is the probability that there is snow?