STATISTICS for Prelim Students

Hilary Term, 2005

These tutorials are designed to provide you with an introduction to basic concepts in statistics, namely the presentation of data, measures of central tendency and dispersion), statistical inference (elementary probability theory and probability distributions, sampling estimation and hypothesis testing), correlation and regression (correlation coefficients, correlation and causality, ordinary least squares regression and statistical inference of simple regression equations.

We will meet for 2 hour long classes in Weeks 1, 2, 3 and 4.The classes for Week 1 are in the Seminar Room, NuffieldCollege and are from 2p.m. till 4 p.m.The tutorial outline and questions for each week will be available at the following website:

For the classes I will ask a few of you to make 5 minute presentations in class on each of the topics listed.Questions will be discussed and some answer solutions will be handed out during class. Some additional questions may be handed out in the tutorial for students to attempt on the board.

Please make sure that you understand the key concepts that you have been asked to make notes on. They are aimed at developing your understanding and will also help you to answer the questions set. Please feel free to bring along the key textbooks. I am going to primarily make use of Wonnacott and Wonnacott and of Aczel (see details on: Both are reasonably good but I prefer Aczel.

Week 1

Reading List

Aczel, Chapters 1, 2

Wonnacott and Wonnacott: Ch 1, 2, 3

Permutations, combinations and basic probability rules

Make notes on the followingand, where appropriate, demonstrate your understanding with the use of an example:
  • Experiment, Random Experiment
  • Event
  • Sample space
  • The Probability of an Event
  • Law of Large Numbers
  • Random Sample
  • Mutually exclusive events
  • Factorial rules
  • Permutations and combinations
  • The multiplicative and additive rules of probability
  • Independent events
  • Complementary events
  • Conditional Probability
  • Bayes Theorem

Note the following fundamental counting rules in probability of multiplication and addition

Addition Rule.

If event A can occur in m possible ways and event B can occur in n possible ways,
there are m+n possible ways for either event Aor event B to occur,
but only if there are no events in common between them.
n(A or B)=n(A)+n(B)-n(AB)

Multiplication Rule (can only be applied if the events are independent)

If event A can occur in m possible ways and event B can occur in n possible ways,
there are m•n possible ways for both events to occur.
n(A and then B)=n(A)×n(B)

Now do the following questions

Q1 Calculate by hand, showing your work, the following combinations:

  1. 5C3.
  2. 6C4.
  3. 9C3.
  4. 10C4.
  5. 9C9 (explain why the answer is one)

Q2 Calculate by hand, showing your work, the following arrangements:

  1. 5P3.
  2. 6P4.
  3. 9P3.
  4. 10P4.
  5. 9P9.

Q3 Assume you have 20 M&M candies as follows: 5 orange, 6 yellow, 5 red, and 4 green. In one selection, how many ways can you select 1 orange or 1 yellow M&M? What is the probability of selecting 1 orange or 1 yellow M&M?

Q4Assume you have 20 M&M's colour distributed as above. If selected without replacement, in how many ways can you select two red ones in two selections? What is the corresponding probability?

Q5 How many different ways (i.e. is it a boy or a girl) can parents have three children?

Q6 You need to study, practice football, fix dinner, phone your parents, and go buy a notebook. How many different ways can you arrange your schedule?

Q7 You have five places left for stamps in your stamp book and you have eight stamps. How many different ways can you select (combine) five? How many different ways can you arrange them (permutation) ?

Q8 How many permutations are possible of the word STATISTICS

Q9 How many permutations are possible with the word TRINITY

Q10 Imelda Marcos has 2003 pairs of shoes. In how many different ways can she select a left shoe then a right shoe?

Q11 Scanti Lee Clad has 15 short shorts and 11 tiny blouses. In how many different ways could she select a shorts-blouse combination?

Q12 The PPE reading list contains 12 Philosophy books and 15 Politics books.

  1. In how many different ways could a student select a Phil or Pol book?
  2. In how many different ways could a student select a Phil and then a Pol book?
  3. In how many different ways could a student select a Pol and then another Pol book?

Q13 In how many ways could you arrange the following?

  1. Four notebook sections from a set of six sections?
  2. Ten homeworks from a set of twelve homeworks?
  3. Five tests from a set of eight tests?
  4. All 20 questions from a set of 20 questions?

Q 14 Six girls start playing a volleyball game.

  1. In how many ways could the six positions be filled?
  2. If the positions are selected at random, what is the probability that Nikki will be server?
  3. Express the probability in part b above as a percentage.

Q 15 In a class of 24 students, 7 are left-handed, and the rest right-handed. If 8 people are selected at random from this group, what is the probability that

  1. 3 are left-handed and 5 are right-handed?
  2. all are right handed?
  3. all are left-handed?

Q16 A box contains 14 identical balls of which 6 are white, 5 red and 3 black. $ balls are drawn. What is the probability that

a.two are red

b.one is white

Q17 A bag contains 6 red &4 white balls. Two balls are drawn at random from the bag. Find the probability that (i) both are white and (ii) one is red and one is white

Q18 A bag contains 4 white and 3 black balls. Another bag contains 5 black and 3 white balls. A ball is drawn from one of the bags selected at random. What is the probability that it is white?

Q19. A can hit a target 4 times in 5 shots, B can hit 3 times in 4 shots and C twice in 3 shots. All three sleep in a dorm. Early one morning a little birdie starts cheeping really loudly. All three fire a volley. What is the probability that at least 2 shots hit (and hopefully kill or permanently disable the cute little bird)?

By using the complementary rule with the multiplication rule, one can find the probability of at least one event being what we want. At least one means the same as one or more. The complement of one or more is none. So the multiplication rule is used to find P(none) and then take the complement of it. P(at least one) = 1 - P(none)!!!

Now attempt the following question.

Q20 A person deals you a new five cards hand. What is the probability of having at least one heart?