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Probability games
In board games, players often roll dice or toss coins to determine the order or number of steps they can take. Luck, and therefore probability, is an essential mechanism in many games.
What do you notice when you roll a dice?
– The possible outcomes are: \latex{ 1, 2, 3, 4, 5 } and \latex{ 6 }.
– It is impossible to roll a number larger than \latex{ 6 }.
– You can't roll a number smaller than \latex{ 1 }.
– You will need a bit of luck to roll a \latex{ 6 }.
– It is impossible to roll a number larger than \latex{ 6 }.
– You can't roll a number smaller than \latex{ 1 }.
– You will need a bit of luck to roll a \latex{ 6 }.
If you toss a coin twice, it is possible that
– both tosses will come up heads (h h);
– the first toss will come up heads, and the second toss will come up tails (h t);
– the first toss will come up tails, and the second one will come up heads (t h);
– both tosses will come up tails (t t).
– the first toss will come up heads, and the second toss will come up tails (h t);
– the first toss will come up tails, and the second one will come up heads (t h);
– both tosses will come up tails (t t).
There are no other possibilities.
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Games with coins
1st game: FRUIT HUNT
Start the game from the circle at the top of the board (\latex{ 1 }). Toss a coin to decide which way to go. Heads mean left, and tails mean right. In three tosses, you'll reach the bottom row (\latex{ 7, 8, 9 }, or \latex{ 10 }). Each circle represents a fruit: a pear, a lemon, some grapes, or an apple.
- Play \latex{ 20 } games in a row and record how many times you arrived at each fruit.
- Choose a fruit and write down how many times you got to it.
- Play the game in pairs. Choose two fruits each (choose different fruits). Take turns to play. You only get a point if you land on one of your chosen fruits. The player with the most points at the end of the \latex{ 10 } games wins.
\latex{1}
\latex{3}
\latex{2}
\latex{4}
\latex{5}
\latex{6}
\latex{7}
\latex{8}
\latex{9}
\latex{10}
\latex{ h }
\latex{ t }
\latex{ h }
\latex{ t }
\latex{ h }
\latex{ t }
\latex{ h }
\latex{ t }
\latex{ h }
\latex{ t }
\latex{ h }
\latex{ t }
2nd game: COIN COMPANIONS
Start the game from the circle at the top of the board (\latex{ 1 }). Toss a coin to decide which way to go.
\latex{ 1 }st variation: Heads mean left, and tails mean right. In two tosses, you'll reach the bottom row (\latex{ 4, 5, 6 }, or \latex{ 7 }). Each circle represents an animal: a bird, a frog, a mouse, or a snail.
- Play \latex{ 20 } games in a row and record how many times you arrived at each animal.
- Choose an animal and write down how many times you got to it out of the \latex{ 20 } games.
- Play the game in pairs. Choose two animals each (choose different animals). Take turns to play. You only get a point if you land on your chosen animals. The player with the most points at the end of the \latex{ 10 } games wins.
\latex{ 1 }
\latex{ h }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ 5 }
\latex{ 6 }
\latex{ 7 }
\latex{ t }
\latex{ h }
\latex{ t }
\latex{ h }
\latex{ t }
\latex{ 2 }nd variation: Same as the previous game, but this time go left if the coin comes up tails and right if it comes up heads.
\latex{ 3 }rd variation: Play the previous game, but roll a die to decide which way to go. If you roll a \latex{ 6 } you go left, if you roll \latex{ 1, 2, 3, 4 } or \latex{ 5 } you go right.
3rd game: CIRCLE BACK
Start the game from the circle at the top of the board (\latex{ 1 }). Toss a coin to decide which way to go. Heads mean left, and tails mean right. Circle \latex{ 3 } is special, because if you toss a tail there, you will have to stay where you are. So it is not certain that you will reach the bottom row (\latex{ 4, 5, 6 }) in two throws.
- Play the game \latex{ 20 } times and record how many steps it took to get to the bottom row.
- Try to guess the number of steps needed to get to the bottom row for each game. Write down how many times you guessed correctly.
- Write down \latex{ 8 } different series of steps that can get you to the bottom row. How many possible series are there in total?
- Which of the following statements are true, possible or impossible?
– You can reach the bottom row in two steps.
– It took \latex{ ht } steps to get to the bottom row.
– It took four steps to get to the bottom row, \latex{ hhht }.
– You can reach the bottom row by taking the following four steps: \latex{ ttth }.
– It took \latex{ ht } steps to get to the bottom row.
– It took four steps to get to the bottom row, \latex{ hhht }.
– You can reach the bottom row by taking the following four steps: \latex{ ttth }.
\latex{ 1 }
\latex{ h }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ 5 }
\latex{ 6 }
\latex{ t }
\latex{ h }
\latex{ t }
\latex{ h }
\latex{ t }
