Take 13 cards of a suite (Ace, 2, 3, 4, ... , King). The game is as follows: you pick two cards, and then the third. If the third card is between the first two cards, you win. What are the chances of your winning?
If you solve the 13 card puzzle, try the same with all 52 cards
A man has two girlfriends, and wants to see them equally often. Trains leave from his station to each of their stations every one hour. So he decides that he is going to land up at the station at a random time, and take whichever train comes first. Fair? But he finds out that he is seeing one girlfriend three times as much as the second. Why?
Answer: The train to first arrives 15 minutes after the train to second
Instructor Notes: The key is for kids to understand that outcomes that look equally likely can have different chances of occurrence.
Monty Hall: 3 doors, Goats behind two and treasure behind the third. Pick - and then the host will reveal one door and ask if you want to switch? Should you switch? (Answer: Yes)
Instructor Notes: Do with props - allow kids to experiment 20 times. Page 81 Chart on CuriosIncident.
Bertrand Paradox: You have three boxes. One has two gold coins, second has two silver coins, third has one gold and one silver coin. You draw a box at random, and then a coin at random - it happens to be a gold coin. What are the chances that the other coin is a gold coin? (Answer: 2/3)
Instructor Notes: This is an intuitively hard problem, and children may reach a conclusion of 1/2. Play the game 20 times to illustrate. While choice of GG and GS boxes is equal, the former has 100% chance of throwing up a gold coin, while latter has 50%.
Joke: (MC - 0 - 11) A teacher drew several circles on a sheet of paper. Then he asked a student "How many circles are there?" "Seven" was the answer. "Correct! So how many circles are there?" the teacher asked another student. "Five," answered the student. "Absolutely right!" replied the teacher. How many circles where there on the sheet?
Answer: 12, 7 on one side and 5 on another - Ask your parents
Contributed by Smiti: There are five boxes with five numbers in them in ascending order. You are given a sixth number - how many boxes do you have to open to see if the sixth number is same as any of the numbers in any of the boxes?
Answer: 3 (Compare with the middle, and then if it is lower, compare with bottom two; if is is higher than middle, compare with top two)
Birthday Problem: 30 people in a class. What is the probability that at least two people share a birthday? (Answer: about 70%. The probability crosses 50% at 23 kids)
MartinShCol - 2.10 - A kid has to play three games of arm wrestling and will earn a reward if he wins two consecutive games out of three. He can choose to play Dad,Mom,Dad or Mom,Dad,Mom. If Mom is a stronger player, which should be choose (Answer: Mom,Dad,Mom)
Homework Problem
Game of Pig: A game works as follows - The first person to score 100 points wins. In each turn, a player can roll two dice as many times as they want. The total on those dice add to their score. However, if at any point, one dice turns up "6", then the score for that chance goes to zero. If both dice turn "6", then their entire score goes to zero. Think about a strategy to win the game. You may use this http://nrich.maths.org/1258 to play.
The Colossal Book of Short Puzzles and Problems, by Martin Gardner
The Curious Incident of the Dog in the Night Time, by Mark Haddon
NOT USED
Equally likely versus not
Probability of two dice roll resulting in number 2 (Answer: 1/36)
Why is it not 1/12 since there are 12 outcomes?For this to work, outcomes have to be equally likely. Else it has to be weighted by likelihood of that outcomeGet kids of take examples where outcomes are not equally likely, and hence straight computation doesn't work
Structure as a game where we shuffle the kings and aces and place them. If the hearts king and aces are together, you win, else you lose. What are the chances of winning? We have Three aces (hearts, spades, diamonds) that are alternated with 4 Kings in placement of 7 cards. What are the chances that the king and ace of hearts are next to each other? (Answer: 1/2)
Instructor Note: Kids can do this with full calculations, or simpler by just seeing probability at any position of Ace
Dependent Events
There are three green and two red marbles in a bag. You win $1 if you pick two green marbles in first two draws. Would you want to play the game for $0.35? Would you play it if the first marble was replaced before the second draw? (Answer: No and Yes)
Shakuntala - 142 above, with Identical twins (Answer: 1/2)
Probability with Combinatorics
There are 9 people including me, and a group of 3 has to be chosen at random. What are the chances that I am part of the chosen group (Answer: 8C2 / 9C3)
Instructor Notes: Students may have to be reminded of nCm notion and formula
Probability with Cases
Take 5 dice - 3 black and 2 red. Lets say the red dice are loaded so that probability of "1" coming up is 1/4 and not 1/6. If you randomly choose a dice and then roll it, what is the probability of "1" coming up? (Answer: 3/5.1/6+2/5.1/4=1/5)
In a game of tennis, you have an opponent who is twice as likely to win a point as you are. Should you play a best-of-1 or best-of-5 game with him? (Answer: 1/3 versus 17/81, so get it over as soon as possible - in long term "skill wins")
Expected Value
In a dice roll, you earn $1 for each even roll, and $3, $9, $15 for 1, 3, 5 respectively. What is the expected value? (Answer: 5.5)
Instructor Notes: Let kids try it, likely to get to 6.5 at first go. Let them think about why that doesnt work. Guide through counting if required.
This is a game of coin flips. You win doubling amounts for each heads roll, but the game ends if you roll a tails. How much would you pay to play the game? (Answer: infinite)
Probability with Geometry
You pick two random numbers between 0 and 1, and if x+2y > 1, you win. What are your chances of winning (Answer: 3/4)
Instructor Notes: Let kids think, and then give a hint around geometry/ plotting. You can find the answer easily by plotting the winning curve on a 2-D graph
Miscellaneous Problems
MartinShCol - 2.14 - In a card game, you are allowed to draw cards till you draw the first black one. You play this game a hundred times. What will be the ratio of red to black cards? (Answer: 1/2)
Brilliant.org: Boris and Alfred run at 10km/hr and 7km/hr respectively in a 70 km race. Boris trips at some random point, hurts himself, and runs the rest of the race at 5km/hr. What is the probability that Alfred wins? (4/7)