Additional revision questions

Suppose that one card is to be selected from a deck of twelve cards that contains six red cards numbered from 1 to 6 and six blue cards numbered from 1 to 6.

1. (a)

   Write down the sample space for the experiment.

2. (b)

   Let $A$ be the event that the number on the card is odd, let $B$ be the even that the card is red, and let $C$ be the event that the number on the card is less than 4. Write down the subset of $\mathrm{\Omega }$ corresponding to each of the events $A$, $B$ and $C$.

3. (c)

   Write down the subsets of $\mathrm{\Omega }$ corresponding to each of the following events, both in terms of intersections, unions, and complements of $A$, $B$ and $C$, and explicitly giving the sample points in each event:

   1. i.

      Both $A$ and $B$ occur but not $C$.

   2. ii.

      At least one of $A$, $B$ and $C$ occur.

   3. iii.

      Exactly one of $A$, $B$ and $C$ occur.

The genetic code specifies an amino acid by a sequence of three nucleotides in order. Each nucleotide can be one of four kinds, $T$, $A$, $C$ or $G$, with repetitions permitted. What is the sample space of possible amino acid codings? If all codes are equally likely to occur, what is the probability that a code contains two $T$’s and a $G$?

A fair dice is tossed seven times. Let $E_r$ be the event that a six occurs exactly $r$ times in the seven throws, and let $p_r=𝖯(E_r)$. Define an appropriate sample space $\mathrm{\Omega }$ (with equally likely sample points) for this experiment. Compute the number of sample points in $\mathrm{\Omega }$ and each $E_r$, and hence find $p_r$ for $r=0,1,\mathrm{\dots },7$.

1. (a)

   Let $\mathrm{\Omega }$ be the sample space, and let $A$ and $B$ be events. Let $C$ denote the event that exactly one of $A$ and $B$ occurs. Draw a Venn diagram for $C$ and write down an expression for $C$ in terms of unions, intersections, and complements involving $A$ and $B$.

2. (b)

   Now let $𝖯$ be a probability measure defined on the events of $\mathrm{\Omega }$. Write down an expression for $𝖯(C)$ in terms of $𝖯(A)$, $𝖯(B)$ and $𝖯(A\cap B)$. Prove your result using the axioms of probability.

3. (c)

   The probability that Alice will fail the test is 0.13. The probability that Bob will fail it is 0.19. The probability that they will both fail is 0.05. Find the probability that

   1. i.

      Exactly one of them fails.

   2. ii.

      At least one of them fails.

   3. iii.

      Both pass.

First Little Piggy goes to market with probability $0.25$ if Second Little Piggy goes to market. Second Little Piggy goes the market with probability 0.5 if First Little Piggy goes to market. Sadly (for the Piggies) at least one of them must go to market. How likely are both of them to go?

This is a puzzle from a weekend newspaper. Solve it by carefully defining two events, and exploring conditional probabilities, and probabilities of the intersection and union of the events.

Aliens are hiding in one of $n$ planets. For $i=1,\mathrm{\dots },n$ the alien is on planet $i$ with probability $p_i>0$. If the aliens are on planet $i$, when you search planet $i$ you discover them with probability ${\alpha }_i$. Consider events $A_i=\{\text{Aliens are on planet }i\}$ and $S_i=\{\text{Search planet }i\text{ and are successful}\}$.

1. (a)

   You searched on planet 3 and failed to discover aliens. What is the probability the aliens are on planet 1?

2. (b)

   Instead, suppose you searched on planet 1 and failed to discover aliens. In this case, what is the probability the aliens are on planet 1?

See also https://en.wikipedia.org/wiki/Bayesian_search_theory.

A book has 2500 letters per page. Each letter has probability ${10}^{-4}$ of being misprinted, independently of all the other letters.

1. (a)

   Use a binomial distribution to find the probabilities of 0, 1 and 2 misprints on a chosen page.

2. (b)

   Compare the probabilities obtained in part (a) with those obtained using a Poisson distribution to approximate the binomial.

Let $X$ be a continuous random variable with cdf given by

Find the probability density function $f_X$ for $X$. Hence find $𝖤[X]$ and $\mathrm{𝗏𝖺𝗋}(X)$.