Question 1: Analyse and answer the following variation of de Mere’s problem:
Two players play a series of games for the “best out of five.” The winner is to receive a prize of $1000. After three games of play, in which the first player had won one game and the other two games, the match was interrupted by an
earthquake. How should the $1000 be divvied up between the two players so as to properly reflect their likelihoods of having won the series? (Assume the each player has a 50% chance of winning any particular game.)
Question 2: Repeat question 1 but this time assume that the first player has only a 10% chance of winning any individual game.
Question 3: 8640 people walk down the following garden path. At each fork, equal numbers of people take each option. Find the number of people that end up in each of the houses A, B, C, and D.
Question 4: Assume that exactly 50% of children born are boys and 50% are girls.
A couple has three children.
a) Draw a tree diagram displaying the possible genders of their three children, b) What are the chances that the couple has three boys?
c) What are the chances that the couple has at least one boy?
d) What are the chances that the couple has exactly two boys?
Suppose that we are now told that their first child was a girl.
e) What are the chances that the other two children are also girls?
f) What are the chances that at least one of their three children is a girl?
Question 5: Billy’s girlfriend has a dimple on her left cheek (there is 1/100 chance that this occurs), blue eyes (there is a 1/100 chance this occurs), and likes math (there is a 1/100 chance that this occurs). He says that his girlfriend is “one in a million.” Is he correct?
Question 6: A card is drawn at random from a deck of 52 playing cards.
a) Describe the sample space if suits are not considered relevant
b) Describe the sample space is suits and numerical value are considered relevant
c) Describe the sample space if the value of the card is considered irrelevant Question 7: A card is drawn at random from a deck of 52 cards. What is the probability of:
f) Drawing any suit except clubs?
g) Drawing the three of clubs or the king of diamonds or any heart?
Question 8: An urn contains 5 red balls, 8 blue balls, and 7 white balls. A ball is selected at random. What are the chances of:
a) selecting a red ball?
b) not selecting a blue ball?
c) selecting a ball that is white or blue?
After the ball is selected, it is returned to the urn, and the experiment is repeated.
What are the chances, in the run of these two experiments, of d) selecting a red ball followed by another red ball?
e) selecting two balls of the same colour?
Question 9:
The chances that someone gets bitten by a dog at least once in life is 0.02 . The chances of being hit by a meteorite at least once in life is 0.001 .
The chances of stepping in gum at least once in life is 0.99 . What is the probability …
a) Of being hit by a meteorite and being bitten by a dog in your life?
b) Of never being hit by a meteorite?
c) Of never stepping in gum and never being hit by a meteorite?
d) Of all three events happening in your life.
e) None of these events happening in your life.
Question 10: M&M’s come in six colors. Here’s a table showing the probability that a randomly chosen M&M has a particular colour:
Colour Brown Red Yellow Green Orange Blue Probability 30% 20% 20% 10% 10%
a) Fill in the missing number for blue.
b) What are the chances that an M&M chosen at random is either brown or red?
c) What are the chances that two M&M’s chosen at random from an extremely large bag are both blue?
d) What are the chances that three M&M’s chosen at random from an extremely large bag are all blue?
I’ve been told that the colour distribution for Peanut M&Ms is different.
Obtain a bag of peanut M&M’s and use your sample to make estimates for the entries in the following table:
Colour Brown Red Yellow Green Orange Blue Probability
Question 11 (ANNOYING – BUT INTERESTING):
It is said that a Friday falls on the 13th day of the month 48 times every 28 years.
a) Verify this calculation.
b) What is the probability that a randomly chosen Friday is a “Friday the 13th”?
Question 12: (HARD-ISH, BUT REALLY INTERESTING!) There are eight possible outcomes in tossing a coin three times:
HHH, HHT, HTH, THH, HTT, THT, TTH, and TTT.
Two players decide to play the following game. Player A chooses the sequence HHH and player B the sequence THH. A coin is tossed repeatedly until one of these sequences appears. For example, the coin might produce T, T, H, T, H, H and player B wins. If the coin produces the sequence H, T, T, H, H, H then player A wins.
a) Play the game 10 times. Does player B seem to win the majority of times?
b) Explain why player B has the advantage.
c) Suppose instead player A chooses the sequence HHT and player B the
sequence THH. Play the game 10 times. Does player B again win the majority of times? Can you explain why?
d) Here’s a table showing all the options A could choose and what B chooses in
Play the game 10 times for each of the eight rows in the table. Verify that B wins the majority of times in each case. Show me the results you obtained.
Question 13: A bag contains a red ball and a white ball. Jodie takes out a ball at random. If it is red she wins. If it is white, she then moves to a bag that contains two red balls, and a single white ball, and pulls out a ball. If it is red, she wins. If it is white, she then moves to a bag that contains three red balls and a single white ball, and pulls out a ball. If it is red, she wins. If it is white, she then moves on to … There are an infinite number of bags available to her, and she keeps playing this game until she eventually wins.
Explain how this hypothetical scenario “proves” the following equation:
1 1 2 1 1 3 1 1 1 4
2 + 2× + × × + × × × +3 2 3 4 2 3 4 5 ⋯=1
(That is, in math notation, we’ve established:
1
Question 14: A bag contains a red ball, a blue ball, and a white ball. Schuyler pulls a ball out at random. If it is red, he wins. If it is blue, he loses. If it is white, then he moves on to a bag that contains two red, two blue and one white ball. He pulls one out at random. If it is red, he wins. If it is blue, he loses. If it is white he moves on to a bag that contains four red, four blue, and one white ball. And so on, with double the number of red balls and double the number of blue balls from bag to bag.
a) Explain why Schuyler’s chances of winning this game are ½.
b) Write an interesting infinite sum based on this hypothetical scenario whose value is ½.
Question 15: Three dice are tossed simultaneously. What are the chances of rolling …
a) three sixes?
b) two sixes and a one?
c) no sixes?
d) at least one six?
Question 16: Three dice and two coins are tossed simultaneously. What are the chances of receiving …
a) three sixes and two heads?
b) no sixes and two heads?
c) no sixes and no heads?
Question 17: Five cards are drawn from a deck of 52 cards:
a) Explain why the chances of pulling out five cards that are all hearts is:
1 12 11 10 9
0.05%
4 51 50× × ×49×48≈ .
b) Find the probability of pulling out five black cards.
c) Show that the probability of pulling out four Kings among the five cards is close to 0.002%.
d) Show that the probability of pulling out three Kings and two Queens is close to 0.001%.
Question 18: Consider the following magic square:
Player A chooses a number at random from the first row; player B chooses a number at random from the second row, and player C chooses a number at random from the third row.
a) What are the chances that player Bs number is higher than player As?
b) What are the chances that player Cs number is higher than player Bs?
c) What are the chances that player As number is higher than player Cs?
[In this game of chance, B has the advantage over A, C has the advantage over B, and A has the advantage over C!]
Question 19:
a) Eleven numbers are arranged in a line. The first number is 0, the last number is 0, and every number in between is the average of its two neighbors. What are the 11 numbers and why?
b) Eleven numbers are arranged in a line. The first number is 0, the last number is 1, and every number in between is the average of its two neighbors. What are the 11 numbers and why?
Question 20: You are a game show contestant and the game show host presents to you 100 boxes. She tells you that inside one box lies a fabulous prize and all the remaining boxes are empty. You select a box at random and are about to open it when the host interrupts you and opens 98 boxes to reveal to you their emptiness.
This leaves two boxes: the one you selected and one other.
You are now given the chance to “stick” with the box you first chose, or to “switch”
and open instead the second box.
a) If you decide to stick, what are your chances of winning the prize?
b) If you decide to switch, what are your chances of winning the game?
Suppose the game show host opens only 97 boxes. This leaves three boxes: the one you first selected and two others.
The host now gives you the choice to either “stick” with your original box or to switch to either one of the remaining boxes.
c) If you decide to stick, what are your chanced of winning the prize?
d) If you switch to a different box, what now are your chances of winning?
Question 21: In a game, if
a
outcomes are deemed “favorable” and the remainingb
possible outcomes “unfavorable,” then folk may say – in horse racing circles in particular – that the odds in favor of winning are “a
tob
”, or alternatively that the odds against are “b
toa
.” For example, in rolling a die the odds in favor of rolling a 6 are 1:5. The odds against rolling a 5 or a 6 are 4:2 (which could be reduced to 2:1). In a horse race if the odds against a horse are 7:2, this means that bookies believe that the horse has only a 29 chance of winning.
CORRECT or INCORRECT?
a) A bookie at a horse race says that the odds against a particular horse are 5:8. This means that the probability the horse will win the race is 8
13.
b) A game yields a 30% chance of a win. The odds against winning the game are thus 7:3.
c) In casting a die, the odds in favor of rolling a number smaller than 5 are 2:1.
d) In tossing a coin twice, the odds against receiving two heads is 3:1.
Question 22:
Bag 1 contains 13 red balls and 14 blue balls.
Bag 2 contains 12 red balls and 7 blue balls.
A bag is selected at random and a ball is pulled out of that bag at random. We are told that the ball is red.
What is the probability that the ball came from bag 1?
Question 23: A die is tossed. What is the probability that the result is a number less than 4 if …
a) We are told no other information?
b) We are told that the result was an odd number?
c) We are told that the result wasn’t 5?
d) We are told that the result wasn’t 1?
Question 24:
a) Two ordinary dice are tossed. What is the probability of NOT getting a total of 7 or 11?
b) Two Sicherman dice are tossed. What is the probability of NOT getting a total of 7 or 11?
Question 25: One bag contains 4 red and 5 white balls. A second bag contains 3 red and 6 white balls. A ball is drawn from each bag. What is the probability that …
a) Both balls are white b) Both balls are red
c) One is white and the other is red
Question 26 One bag contains 2 red and 3 white balls. A second bag contains 3 red and 1 white balls. A ball is drawn from each bag. Suppose we are told that one ball chosen was red. What are the chances that the second ball is also red?
Question 27: A bag contains 5 red and 4 white balls. A ball is selected and then, without replacing the first ball, a second ball is selected. I tell you that the second ball is white. What is the probability that the first ball was white?
Question 28: You play a simple coin-tossing game. If the coin lands heads, you win
$3. If lands tails, you must pay $1.
a) If you play this game 100 times, how much money do you expect to have?
b) What is the expected value of this simple game?
Question 29: Roll a die. If it comes up even, you win that many dollars. If it comes up odd, you must pay that many dollars. (For example, a roll of “4” wins you four dollars. With a roll of “5,” you lose five dollars.)
What is the expected value of this game? Would you want to play it?
Question 30: A coin is tossed once, possibly twice. If a head appears on the first toss, you win $10 and the game stops. If it lands tails, the coin is tossed again. If the second toss lands heads you win $4, otherwise you pay $20.
a) If you played this game 100, on average, how many times will you win $10?
How many times will you win $4? How many times will you lose?
b) What is the expected value of this game? Would you want to play it?
Question 31: A gambling game is called “fair” if its expected value is zero.
A die is rolled. If it rolls 1, 2, 3, or 4, you win $300. If it rolls 5 or 6 you lose $x.
Find a value of x that makes this game fair.
Question 32: A gambling game is called “fair” if its expected value is zero.
A coin is tossed three times. If at least two heads appear, you win $100. If exactly one head appears, you win $50. If no head appears, you lose $x.
Find a value of x that makes this game fair.
Question 33: A die is rolled. If it lands 1 you win $10. If it lands 2 you win $300.
If it lands 3 you win $1. If it lands 4 you lose $500. If it lands 5 or 6 you win $x.
Find a value of x so that the expected value of this game is fifty cents.
Question 34: There is a one-in-twenty-million chance of winning the lottery.
This week’s jackpot is $50,000,000. If it costs $1 to buy a ticket, what is the expected value of this game? Are the odds in your favour?
[ASIDE: They are! But what fact of social behaviour are we ignoring in this
argument? Why should one still not bother to buy a lottery ticket even if the prize is so high so as to give the impression that odds are in your favor.]
Question 35: PSEUDO-RANDOM NUMBERS
It is not possible to generate truly random numbers with a computer - any program follows a predetermined set of instructions – but it is possible to create a list that appears to be random. Several methods for doing so exist. The most popular is the
“middle-square method” developed in 1946 by John von Neumann. It works as follows:
Step 1: Select a four-digit number.
Step 2: Square the number to produce an eight-digit number. (You might have to place a zero at the front of the number to get eight digits.) Step 3: Use the middle four digits of this eight-digit number as the next number in the sequence.
REPEAT
This procedure produces a seemingly random list of numbers between 0 and 9999.
a) Verify that starting with the number 7254 yields the sequence:
7254, 6205, 5020, 2004, 0160, 0256, 0655, 4290
b) What happens if, instead, you start with the initial number 1049?
This procedure (and, in fact, all procedures that currently exist) are not without flaw.